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Reflected diffusion, no-flux continuity equations and confined Lagrangian flows in bounded domains

REFLECTED DIFFUSIONS, NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS IN BOUNDED DOMAINS

arXiv:2607.28344v1 [math.CA] 30 Jul 2026

RAMA CONT

Abstract. Motivated by marginal distribution flows of reflected diffusions in bounded domains, we investigate when a density–flux pair solving a no-flux continuity equation admits a regular Lagrangian flow that remains in the closed domain and generates the prescribed density flow. Our first result gives sufficient conditions in terms of interior bounded-variation regularity, bounded-variation control on a boundary collar, a one-sided bound on an absolutely continuous divergence, and vanishing normal trace of the velocity. The proof uses the fact that tangency removes the singular boundary contribution to the divergence of the zero extension, thereby making the extended velocity admissible for the Ambrosio–DiPerna–Lions theory. We also establish two uniqueness results for no-flux Fokker–Planck equations: a duality result for bounded measurable drifts and a weighted-energy result for entrance-type drifts singular at the boundary. Our second result shows that these boundary assumptions cannot be jointly relaxed so as to admit a boundary-current mechanism. We construct an explicit smooth density–flux pair carrying a boundary current—a tangential mass current of nonvanishing linear density along a wall where the volume density vanishes. Its density evolution is unique in a weighted class, and its characteristics are unique, confined, and transport the marginals, yet it admits no regular Lagrangian flow because the compressibility bound fails arbitrarily close to the initial time. Rigidity results delimit such failures and show that the relevant boundary hypotheses are structurally entangled. As an application, we provide precise regularity assumptions under which the reflection-free probability-flow ODE describes the flow of marginals of reflected diffusion models, after early stopping. Our results provide a rigorous mathematical justification for using the ODE-based sampling of reflected diffusion models under minimal regularity assumptions on the coefficients, and also indicate when such ODE-based samplers may fail.

Keywords: Score-based diffusion models; Reflected diffusion processes; Fokker–Planck equations; confined Lagrangian flows; generative models DiPerna–Lions theory; Continuity equation; Linear transport equations.

Mathematics Subject Classification: 34A12 35D30 35Q84 35Q49 60H10 49J52 35J60 35Kxx 28A25

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Contents 1. Generative models, reflected diffusions and density flows in domains 1.1. Generative modelling with reflected diffusions 1.2. Probability flows and regular Lagrangian flows in bounded domains 1.3. Main results 1.4. Sharpness and rigidity 1.5. A mimicking theorem for reflected diffusions 1.6. A uniqueness theorem for Fokker-Planck equations with singular drift at the boundary 1.7. Outline 2. Definitions and preliminary results 3. Confined regular Lagrangian flow 3.1. Boundary sheet identity 3.2. Assumptions and proof of Theorem A 3.3. Tangency and time reversal 4. The boundary current 4.1. Construction of the boundary current 4.2. Weighted uniqueness for no-flux Fokker–Planck equations with entrance-type singular drifts 4.3. Eulerian perspective: uniqueness of the density evolution 4.4. Characteristics, marginal transport, and failure of compressibility 4.5. Numerical illustration 5. Mimicking reflected diffusions with differential equations 5.1. The reflection term disappears from the probability flow 5.2. Failure at the endpoint: the boundary is innocent 6. Rigidity: constraints on boundary mechanisms 7. Discussion and extensions Acknowledgements References

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1. Generative models, reflected diffusions and density flows in domains 1.1. Generative modelling with reflected diffusions. Reflected diffusion models for constrained generative modelling were introduced by Lou and Ermon [29] and Fishman et al. [18] to address the problem that standard score-based diffusion models [33] can generate unnatural samples outside the support of the target data distribution. In this setting, the forward process is constrained to remain in a bounded domain Ω ⊂ Rd representing the data constraints: it evolves as a reflected diffusion process in Ω , starting from the data distribution µ0 , supported on Ω: (1)

dXt = f (Xt , t) dt + σ(t) dWt − ν(Xt ) dLt ,

Xt ∈ Ω,

X0 ∼ µ0 ∈ P(Ω)

where σ : [0, T ] → R+ is a scalar function, Lt is the local time of X at the boundary ∂Ω of the domain, which reflects the process into the interior whenever it reaches ∂Ω, and ν(x) denotes the outward unit normal vector to the boundary at x ∈ ∂Ω. This forward diffusion transports the data distribution µ0 (which may have a singular support) towards a (tractable) target distribution µT ∈ P(Ω) with density pT . Sampling is then done by reversing the density flow (pt )t∈[0,T ] of (1). This can be done by simulating the time-reversed reflected SDE [7, 30]
dYt = − f (Yt , T −t)−σ(T −t)2 ∇x log pT −t (Yt ) dt+σ(T −t) dB t −ν(Yt )dLt , Yt ∈ Ω, Y0 ∼ pT involving the score function ∇x log pt of X, which is learned via score matching [29, 18, 33, 34]. Then the marginal density Yt is qt = pT −t so YT ∼ µ0 gives a sample from the data distribution. Alternatively, as noted by Lou & Ermon [29], the same probability flow (pt )t∈[0,T ] may be obtained by simulating a deterministic probability flow ODE (PF-ODE) (2)

Żt = f (Zt , t) −

σ(t)2 ∇ log pt (Zt ), 2

Z0 ∼ µ0

driven by the velocity field v(x, t) = f (x, t) − 21 σ 2 (t)∇ log pt (x). Simulating this ODE backwards in time starting from ZT ∼ pT then yields the backward flow pT −t , which enables fast deterministic sampling of µ0 and exact likelihood evaluation. These methods yield state-ofthe-art generative models, achieving competitive performance on benchmark data sets such as CIFAR-10 and ImageNet [29]. At the same time, their theoretical foundations remain to be fully explored [8]. The claim [29] that the marginal flows of the ODE (2) and the reflected SDE (1) have the same density flow (pt )t∈[0,T ] appears somewhat surprising at first glance. Interestingly, as noted by Lou and Ermon [29], the probability flow ODE (2) is the same as in the unconstrained case where Ω = Rd [10, 33] and, unlike the SDE (1), does not contain any reflection term. The density flow associated with the reflected SDE (1) satisfies a parabolic FokkerPlanck equation in a bounded domain with a no-flux boundary condition, while the flow associated with the ODE (2) is a first-order transport equation. In fact, if pt > 0 is regular enough, using the identity ∆pt = ∇ · (pt ∇ log pt ) the continuity equation for the vector field v(x, t) = f (x, t) − 12 σ 2 (t)∇ log pt (x) ∂t p + ∇· (p v) = ∂t p + ∇· (p f ) −

σ(t)2 ∆pt 2

formally coincides with the Fokker-Planck equation for (1)! Hence, if uniqueness of solutions for both equations is satisfied, one therefore expects the reflected diffusion and the PF-ODE

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to possess the same flow of marginal distributions (pt )t∈[0,T ) and (2) may indeed be used to sample from pt for t > 0. Needless to say, the above assertions assume many regularity, existence, uniqueness, and positivity properties which deserve to be qualified. Our goal in this study is to give mathematically precise conditions for the above assertions to hold in a general setting. Note in particular that the ODE (2) itself carries no boundary condition. Rather, the associated flux J(x, t) = pt (x)v(x, t) satisfies pt (x)v(x, t) · ν(x) = 0 on the boundary. One of the mathematical questions we address is whether this ’Eulerian’ condition actually produces a regular Lagrangian flow confined to Ω. We will provide conditions for this to hold. Alternatively, we will investigate how deterministic transport can fail when the density degenerates at the boundary. 1.2. Probability flows and regular Lagrangian flows in bounded domains. Motivated by these considerations, we investigate the following question: When does a probability flow over a bounded domain admit a deterministic transport representation? More precisely, given a probability flow (µt )t∈[0,T ] on Ω with possibly singular initial law µ0 , we seek to construct a vector field v satisfying ’minimal’ regularity conditions, which generates a unique regular Lagrangian flow satisfying a no-flux condition at the boundary and transports µ0 along the prescribed flow. We address these questions in a general setting, for general density–flux pairs solving a noflux continuity equation. We identify a BV normal-trace criterion that yields a confined regular Lagrangian flow through the Ambrosio–DiPerna–Lions theory, apply it to early-stopped reflected diffusions, and construct a boundary-current example in which invariant characteristics and exact marginal transport persist while regular-Lagrangian compressibility fails. Let Ω ⊂ Rd be a bounded Lipschitz domain with outer unit normal ν. The object of study is a density–flux pair (p, J) on Ω̄ × [0, T ], say p and J continuous on Ω̄ and continuously differentiable in Ω, satisfying the continuity equation together with a no-flux boundary condition:  ∂ p(x, t) + ∇· J(x, t) = 0, (x, t) ∈ Ω × (0, T ),   t J(x, t) · ν(x) = 0, (x, t) ∈ ∂Ω × (0, T ), (3)   p(·, 0) = p0 in Ω. The boundary condition J(x, t) · ν(x) = 0 means that no mass flows across the boundary ∂Ω: formally, Z Z d pt = − J · ν dHd−1 = 0 dt Ω

∂Ω

so total mass in Ω is conserved and the evolution is confined to Ω̄. When the flux has the Fokker–Planck
form J = f p − 21 σ 2 ∇p, the second line of (3) is the co-normal condition 1 2 2 2 σ ∇p − f p · ν = 0, a Robin-type relation ∂ν p = (2f · ν/σ ) p which reduces to the homogeneous Neumann condition only where f ·ν = 0. The classical formulation (3), however, presupposes that J has a well-defined normal trace on ∂Ω, which fails for the irregular velocity fields we would like to consider. We therefore consider weak solutions of (3) by testing against functions unrestricted at the boundary: a density–flux pair (p, J) is a weak solution of the

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continuity equation (3) in Ω with no-flux boundary conditions if Z Z TZ
∞ p0 φ(·, 0) dx = 0. p ∂t φ + J · ∇φ dx dt + (4) ∀φ ∈ C (Ω̄ × [0, T )), 0

Ω

Ω

This formulation contains simultaneously the interior equation ∂t p + ∇ · J = 0 and the vanishing of the normal flux J · ν on ∂Ω, with no boundary term discarded. Two distinct problems are attached to such a pair, in the terminology of continuum mechanics: the Eulerian description records the evolution of fields at fixed points of the domain, while the Lagrangian description follows individual trajectories, and for smooth velocities each determines the other. The Eulerian problem asks whether the density evolution (pt ) is uniquely determined in a natural class; the Lagrangian problem asks whether the velocity field v := J/p generates a deterministic flow confined to Ω̄ transporting p0 Ld onto pt Ld –a regular Lagrangian flow (RLF) in the sense of DiPerna–Lions [16]. In the whole space, the relation between these two descriptions was analysed in [10] for the probability flows arising in score-based diffusion models: the density flow is unique under weak assumptions on the drift [10, Theorem 3.1], the probability-flow ODE is well posed under Sobolev or BV regularity of the score together with one-sided divergence bounds [10, Theorem 4.5], and a counterexample was used to illustrate this gap. On a bounded domain two new phenomena enter: • the behaviour of v on the boundary ∂Ω, and • the question of invariance — the ODE carries no boundary condition, so nothing in its formulation prevents trajectories from leaving Ω̄. Not surprisingly, the zero-extension of the velocity field v will play a role regarding this point. 1.3. Main results. Our two main results identify, respectively, a sufficient boundary criterion for the transport theorem (in fact the exact criterion eliminating the singular boundary term in the zero-extension approach) and the failure mode when the boundary hypotheses are relaxed. Both revolve around a single identity. For v ∈ L1t BV near ∂Ω, the extension of v by zero, v 0 := v 1Ω , is a BV field on Rd whose distributional divergence is
D· v 0 = (∇· v) Ld ⌞Ω − tr v · ν Hd−1 ⌞∂Ω. (5) There is a discontinuity at the boundary and the gradient has a singular bounded-variation (BV) part. Ambrosio’s theory [1, 2] tolerates bounded-variation singularities, but the divergence acquires a singular sheet carried by ∂Ω unless the normal trace of v vanishes. Tangency of v is exactly the condition removing the singular boundary part from (5), and hence exactly the condition making v 0 admissible for the well-posedness conditions of Ambrosio [1]. Our first main result converts this observation into a transport theorem. Theorem A (Confined regular Lagrangian flow; Section 3). Let (p, J) be a no-flux weak solution on a bounded Lipschitz domain satisfying the flux–density hypotheses (F0)–(F4)
1 [0, T ]; BV (Ω) together with v ∈ of Section 3: positivity of p in Ω; v = J/p ∈ L loc
L1 [0, T ]; BV (Ω ∩ U ) for a neighbourhood U of ∂Ω; absolutely continuous divergence with [∇ · v]− ∈ L1t L∞ (Ω); and tangency, tr v · ν = 0 Hd−1 -a.e. on ∂Ω. Then v admits a regular Lagrangian flow Z on Ω̄, unique up to Ld -null sets, with compressibility constant

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RT exp 0 ∥[∇· v]− ∥L∞ (Ω) , and transporting p0 to pt :
Z(t, ·)# p0 Ld ⌞Ω = pt Ld ⌞Ω

for every t ∈ [0, T ].

Given (5), the proof is short: invariance of Ω̄ is elementary (trajectories of v 0 that exited would be frozen outside, contradicting continuity at the exit time), and the transport identity is read off from (4), which is precisely the statement that the zero extension of p solves the global continuity equation. Tangency is automatic whenever the density is continuous and positive up to the boundary (Corollary 3.3), which covers the marginal flows of reflected diffusions after early stopping. Our second result shows that the boundary assumptions are not merely an artefact of the extension method: they cannot be removed so as to admit the boundary-current mechanism. The construction exhibits a vector field carrying a wall-localised mechanism of the same nature as the singular sheet in (5); the precise relation is an exhaustion statement, given in Section 4. The mechanism is an infinite compression at the wall, and it destroys the regular Lagrangian flow even when everything else — Eulerian uniqueness, invariant characteristics, pointwise transport — survives. Theorem B (The boundary current; Section 4). There is an explicit pair (p, J) ∈ C ∞ (Ω̄ × [0, T ]) on the periodic strip Ω = T×(0, 1) — given in (12) below — with p > 0 in Ω, vanishing linearly at the walls, solving the no-flux continuity equation, such that: (i) (pt ) is the unique no-flux weak solution, with initial condition p0 , of an associated Fokker–Planck equation with uniformly elliptic diffusion, in the relative energy class Cp of Section 4.2; (ii) the normal component v · ν extends continuously to Ω̄ with boundary values that are nonzero almost everywhere, and outward on a moving half of each wall: tangency fails in the classical sense. The tangential component blows up like the inverse distance to the wall, so that v ∈ / L1 (collar) and hypotheses (F2)–(F3) fail there as well; (iii) through a.e. point of Ω there is a unique integral curve of v; it is global, confined to Ω, and the flow so defined transports p0 L2 onto pt L2 for every t; (iv) and yet, for every T > 0, v admits no regular Lagrangian flow on [0, T ]: for any map satisfying the trajectory condition and any constant L, there are a time t∗ ∈ (0, T ] and a set of positive Lebesgue measure on which the push-forward Z(t∗ , ·)# L2 has density exceeding L — the density blowing up like the inverse distance to the wall — so that no single compressibility constant can hold at all times. The mechanism is a boundary current: a tangential mass current of nonvanishing linear density along a wall where the volume density vanishes. The normal flux J · ν vanishes because the density does — not because the velocity is tangential — and near-wall trajectories, swept along the current, are compressed against mid-domain ones without bound. Theorem B separates three layers that the whole-space theory tends to fuse: existence of invariant characteristics, pointwise deterministic transport of the marginals, and the regular Lagrangian flow property. 1.4. Sharpness and rigidity. To investigate the sharpness of these results, we study some (counter)examples. Theorem B shows that the collection of boundary hypotheses (F2)– (F4) cannot be relaxed so as to admit the boundary-current mechanism, the three being structurally entangled in the smooth linear-vacuum regime, so that no example there can isolate (F4); see Remark 4.9. We also show in Section 5, using a reflected Brownian motion on [−2, 2] started from the uniform law on [−1, 1], that the no-flux evolution may be unique in

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the energy class, yet no regular Lagrangian flow may exist from time zero. Here the interior, time-integrated part of hypothesis (F2) fails while tangency holds at all positive times and the boundary points carry proper integral curves: the reflecting boundary is ’innocent’, and on a bounded domain the exclusion argument acquires an instructive twist: the extreme quantile levels are attained at the walls, and trajectories from the vacuum are excluded because they would have to jump there instantaneously. Three rigidity results (Section 6) delimit the possible failure modes and show that the boundary current of Theorem B is the only boundary mechanism available under the stated boundedness and nondegeneracy assumptions — they do not classify every low-regularity boundary pathology in higher dimension. In dimension one, invariance can never fail, mass conservation and no-flux pinning the monotone rearrangement inside the interval; in any dimension, integrable total flux guarantees a superposition of integral curves confined to Ω̄, so no counterexample can consist in trajectories being unavailable; and a Taylor expansion argument shows tangency is automatic wherever the velocity is bounded near the wall, forcing the 1/dist tangential blow-up that the construction of Theorem B realises. 1.5. A mimicking theorem for reflected diffusions. As noted in Section 1.1, the motivation for our study comes from generative models based on reflected diffusions [29, 18], where the probability flow is generated by a diffusion with normal reflection [29] and sampling proceeds by reversing its density flow. For the reflected SDE dXt = f (Xt , t) dt + σ(t) dWt − ν(Xt ) dLt ,

Xt ∈ Ω̄,

the marginal densities solve the Fokker–Planck equation with co-normal no-flux conditions, and the probability-flow velocity is v = f − 21 σ 2 ∇ log pt , with flux J = pt v. A structural observation (Section 5) is that the boundary local time L, which is a stochastic process, is not present in v: the no-flux condition reads pt v·ν = 0, so wherever the density is positive and differentiable up to the boundary the field is tangential and the probability-flow ODE requires no reflection term, in contrast with the reverse SDE, which retains one [7, 30]. Theorem A then applies: after early stopping, for arbitrary — possibly singular — initial laws and for drifts of parabolic Hölder regularity f ∈ C 1+α,(1+α)/2 , boundary Schauder theory for the conormal problem makes the density C 2,α and positive up to Ω̄, tangency holds pointwise, and the early-stopped deterministic reflected sampler is justified exactly as in the whole-space theory of [10], whose Lagrangian theorem likewise imposes joint drift–score hypotheses. This result, which extends the result of [10] to the case of reflected processes, may be viewed as a “mimicking theorem” for the density flow reflected diffusions, in the spirit of Gyöngy [19]. The assumption f ∈ C 1+α,(1+α)/2 is enough to make the zeroth-order coefficient ∇· f of the nondivergence form Hölder-continuous, which is what the estimate requires, though Hölder bounds could equally be imposed on f and ∇· f directly. For merely bounded measurable drifts the density remains Hölder continuous and positive and confined superposition solutions exist, but no mimicking property is claimed. The sharpness results then say precisely what can go wrong at the endpoints: the data mechanism at t = 0 and, for velocity fields not arising from uniformly elliptic diffusions with bounded drift, the boundary current at positive times. 1.6. A uniqueness theorem for Fokker-Planck equations with singular drift at the boundary. The Eulerian side of Theorem B requires uniqueness for a Fokker–Planck equation whose drift has an entrance-type singularity f ·νin ≍ 21 dist(x, ∂Ω)−1 at the boundary, with [∇·f ]− ≍ dist−2 unbounded: the energy method behind [10, Theorem 3.1]RRis inapplicable. Section 4.2 establishes uniqueness in a relative energy class {q : q/p ∈ L∞ , p|∇(q/p)|2 < ∞} by a weighted energy method whose key feature is an exact cancellation: testing the

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equation for the ratio q/p against itself in L2 (p dx), the transport terms produced by the flux cancel against the time derivative of the weight, because p solves the same continuity equation; the singular drift and its divergence never appear, and the boundary is invisible to the energy because the weight vanishes there. We prove this under an assumption (W) independent of the example, as the uniqueness result for no-flux Fokker–Planck equations with entrance-type singular drifts may be of independent interest, as it does not seem to be covered by the encyclopaedic treatment of Fokker-Planck equations by Le Bris and Lions [26]. 1.7. Outline. Section 2 fixes the setting and recalls some key results from DiPerna–Lions– Ambrosio theory. Section 3 gives the proof of Theorem A. Section 4 constructs the boundary current, states a uniqueness result for Fokker-Planck equations with no-flux boundary condition (Section 4.2) and proves Theorem B. Section 5 develops the application to reflected diffusions and the data-endpoint sharpness result. Section 6 proves the rigidity results and discusses the two mechanisms. Section 7 discusses some implications of our results for constrained generative models and some further questions of interest. 2. Definitions and preliminary results Let T > 0, d ≥ 1, Ld be the Lebesgue measure, Hd−1 the (d − 1)-dimensional Hausdorff measure, and Ω ⊂ Rd a bounded domain with Lipschitz boundary and outer unit normal ν. Some statements require ∂Ω ∈ C 1,1 . We also use the flat periodic strip T × (0, 1), T = R/2πZ, whose two boundary circles play the role of ∂Ω; all results transcribe to the annulus {1 < |x| < 2} ⊂ R2 in polar coordinates. Definition 2.1 (No-flux weak solution). A pair (p, J) with p ∈ L∞ ([0, T ]; L1 ∩ L∞ (Ω)), R pt ≥ 0, Ω pt = 1, t 7→ pt Ld narrowly continuous, and J ∈ L1 ([0, T ] × Ω; Rd ), is a no-flux weak solution with initial condition p0 if (4) holds for every φ ∈ C ∞ (Ω̄ × [0, T )). We refer to (4) as (NF). Definition 2.2 (Regular Lagrangian flow [16]). Z : [0, T ] × Rd → Rd is a regular Lagrangian flow (RLF) for a velocity field w if (i) for Ld -a.e. x, t 7→ Z(t, x) is absolutely continuous with Rt Z(t, x) = x + 0 w(Z(s, x), s) ds; and (ii) Z(t, ·)# Ld ≤ LLd for some constant L ≥ 1 and all t. Theorem 2.3 (Ambrosio; DiPerna–Lions [1, 2, 16]). Let w : Rd × (0, T ) → Rd satisfy
(R1) w ∈ L1 (0, T ); BVloc (Rd ; Rd ) ;
(R2) D· wt = (∇· wt )Ld for a.e. t,
and [∇· w]− ∈ L
1 (0, T ); L∞ (Rd ) ; (R3) |w|/(1 + |x|) ∈ L1 (0, T ); L1 + L1 (0, T ); L∞ . Then: (1) for every u0 ∈ L1 ∩ L∞ the continuity equation ∂t u + ∇· (wu) = 0, u(·, 0) = u0 , has at most one bounded distributional solution in L∞ ((0, T ); L1 ∩ L∞ ); (2) there is
an RLF Rt Z for w, unique up to Ld -null sets, with Z(t, ·)# Ld ≤ exp 0 ∥[∇· w(·, s)]− ∥L∞ ds Ld ; (3) for u0 ≥ 0 the unique bounded solution is ut Ld = Z(t, ·)# (u0 Ld ). Theorem 2.4 (Superposition [4, 2]). If t 7→ µt is narrowly continuous, solves ∂t µt + ∇ ·
R T R |w| dµt dt < ∞, then there is η ∈ P C([0, T ]; Rd ) concentrated (wµt ) = 0 in Rd , and 0 1+|x| on absolutely continuous integral curves of w with (et )# η = µt for all t, where et (γ) := γ(t). We shall use repeatedly the one-dimensional quantile identity (see also [10], where it produces closed-form solutions of the probability-flow ODE).

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Lemma 2.5 (Quantile flow). Let d = 1, Ω = (a, b), and let (p, J) solveRthe continuity equation x with pt > 0 a.e., pt v ∈ L1loc , and vanishing flux at a. Set Ft (x) := a pt . Then ∂t Ft (x) = −pt (x)v(x, t), and t 7→ Ft (Zt ) is constant along every absolutely continuous solution of Żt = v(Zt , t); whenever the right-hand side is defined, Z(t, x) = Ft−1 (F0 (x)). Proof. Integrating ∂t p + ∂x (pv) = 0 over (a, x) and using the vanishing flux at a gives the d first identity; then along a solution, dt Ft (Zt ) = ∂t Ft (Zt ) + pt (Zt )Żt = 0, and inverting the strictly increasing Ft gives the formula. □ Finally we state the Eulerian uniqueness result in the natural energy class, the boundeddomain counterpart of [10, Theorem 3.1]. Proposition 2.6 (Uniqueness in the energy class, by duality). Let Ω be a bounded Lipschitz domain, let σ be measurable with 0 < ϵ ≤ σ(t) ≤ ϵ−1 , and let
f ∈ L∞ Ω × (0, T ); Rd . No hypothesis is made on ∇· f , which may be a measure with a nonzero singular part, and none on the boundary behaviour of f . Then in n
o
XΩ := p ∈ L∞ [0, T ]; L1 ∩ L∞ (Ω) : ∇p ∈ L2 [0, T ]; L2 (Ω) there is at most one no-flux weak solution of (6)

∂t p + ∇· (f p) − 12 σ(t)2 ∆p = 0,

1 2 2 σ ∇p − f p

· ν = 0 on ∂Ω,

with a given initial condition p0 ∈ L1 ∩ L∞ (Ω). The proof is by duality against the backward Kolmogorov equation, which for a reflected diffusion carries the Neumann condition ∂ν φ = 0, the adjoint of the co-normal no-flux condition on the forward equation. The mechanism is an exact cancellation rather than an estimate: the drift appears once from the primal equation and once from the dual, with opposite signs, and is never integrated by parts! This is precisely why no condition on ∇· f , and no normal trace of f , is required. Proof. Write a(t) := 21 σ(t)2 ∈ [ 12 ϵ2 , 21 ϵ−2 ], let p1 , p2 ∈ XΩ be no-flux weak solutions with the same initial condition, and set w := p1 − p2 , Jw := f w − a∇w. Since Ω is bounded, w ∈ L∞ ([0, T ]; L2 (Ω)), and Jw ∈ L2 (Ω × (0, T ); Rd ) because f ∈ L∞ and ∇w ∈ L2 . Step 1: the primal equation in variational form. Taking φ(x, t) = χ(x)η(t) with χ ∈ C ∞ (Ω̄) and η ∈ Cc∞ (0, T ) in (4), differenced for the two solutions, gives ∂t w = −∇· Jw in the sense that, for a.e. t, Z (7) ∂t w(t), χ = Jw (t) · ∇χ dx for all χ ∈ C ∞ (Ω̄), Ω ∞ and hence, by density of C (Ω̄) in H 1 (Ω) on a Lipschitz domain and |⟨∂t w, χ⟩| ≤ ∥Jw ∥L2 ∥∇χ∥L2 ,
for all χ ∈ H 1 (Ω), with ∂t w ∈ L2 0, T ; H 1 (Ω)′ . Note that (7) holds for test functions un-

restricted at the boundary: this is the no-flux condition, and it is the only place where it is used. In particular w ∈ C([0, T ]; L2 (Ω)) with w(0) = 0. Step 2: the dual problem. Fix τ ∈ (0, T ] and ψ ∈ L2 (Ω), and consider the backward Kolmogorov
problem with Neumann boundary condition, in variational form: find
φ ∈ L2 0, τ ; H 1 (Ω) with ∂t φ ∈ L2 0, τ ; H 1 (Ω)′ and φ(τ ) = ψ such that, for a.e. t ∈ (0, τ ), Z Z
(8) ∂t φ(t), χ − a(t) ∇φ · ∇χ + f · ∇φ χ = 0 for all χ ∈ H 1 (Ω). Ω

Ω

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Reversing time, φ̃(t) := φ(τ − t) solves a forward problem governed by the family of bilinear forms Z Z
f˜ · ∇u χ on H 1 (Ω) × H 1 (Ω), a(t; u, χ) := ã(t) ∇u · ∇χ − Ω

Ω

which are bounded uniformly in t, measurable in t, and satisfy a Gårding inequality: by Young’s inequality, a(t; u, u) ≥ ã∥∇u∥2L2 − ∥f ∥L∞ ∥∇u∥L2 ∥u∥L2 ≥

2

∥f ∥L∞ 2 ϵ2 ∥u∥2L2 . 4 ∥∇u∥L2 − ϵ2

Lions’ theorem for nonautonomous variational evolution problems [28] therefore provides a unique φ̃, with φ̃ ∈ C([0, τ ]; L2 (Ω)) and φ̃(0) = ψ; undoing the time reversal gives φ. Only boundedness and measurability of f and a are used; the Neumann condition is natural in (8) and requires no regularity of ∂Ω beyond Lipschitz, and no H 2 theory is invoked. Step 3: the exact cancellation. Both w and φ lie in L2 (0, τ ; H 1 ) with time derivatives in R 2 L (0, τ ; (H 1 )′ ), so the Lions–Magenes integration-by-parts lemma [35] applies: t 7→ Ω wφ is absolutely continuous on [0, τ ] and Z Z Z τ  w(τ )φ(τ ) − w(0)φ(0) = ∂t w, φ + ∂t φ, w dt. Ω

Ω

0

Insert χ = φ(t) ∈ H 1 (Ω) in (7) and χ = w(t) ∈ H 1 (Ω) in (8): Z w f · ∇φ − a

∂t w, φ = Ω

Z

Z ∇w · ∇φ, Ω

Z ∇φ · ∇w −

∂t φ, w = a Ω

f · ∇φ w.

Ω

The two drift terms are the same integral with opposite signs, and the two diffusion terms likewise; the sum vanishes identically for a.e. t. Observe that f has been paired with ∇φ throughout and never differentiated: neither ∇· f nor a boundary trace of f appears at any point. R R Step 4: conclusion. Hence Ω w(τ )ψ = Ω w(0)φ(0) = 0, since w(0) = 0. As ψ ∈ L2 (Ω) and τ ∈ (0, T ] were arbitrary, w ≡ 0. □ Remark 2.7 (Duality versus the energy method). The natural alternative is the energy method used in the whole space in [10, Theorem 3.1]: test the equation for w against w R d 1 2 + a∥∇w∥2 = itself, obtaining dt ∥w∥ f w · ∇w. On a bounded domain this route 2 Ω L2 L2 costs two hypotheses that duality avoids. Since w∇w = 12 ∇(w2 ), the drift term must be integrated by parts, which requires D · f to be absolutely continuous with [∇· f ]− ∈ L1t L∞ R and produces, in addition, the boundary term 21 ∂Ω (f · ν)w2 — with no counterpart in the whole space. That term is not annihilated by the no-flux condition, which constrains the total flux a∇p − f p and not the drift flux f p separately; it must either be given a sign (f · ν ≤ 0) or absorbed through a multiplicative trace inequality, at the price of assuming a normal trace with ∥f · ν∥L∞ (∂Ω) ∈ L2 (0, T ). Neither method dominates the other: the energy method tolerates unbounded f , while duality requires f ∈ L∞ in order to
solve the dual problem. Both, however, need global square integrability, f ∈ L2 (0, T ) × Ω , and not merely f ∈ L2loc (Ω): each argument tests
the equation with a function in H 1 (Ω) and so requires Jw = f w − a∇w ∈ L2 (0, T ) × Ω , hence f w ∈ L2 up to ∂Ω, and local integrability inside Ω does not control the behaviour of f as the boundary is approached. Under f ∈ L∞ this is automatic; it is the minimal hypothesis under which the class of admissible solutions is closed under the duality pairing. On a bounded domain, however, the drifts of interest are bounded, and duality is the appropriate default.

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

11

Remark 2.8 (A singular divergence does not obstruct uniqueness). Proposition 2.6 covers drifts whose distributional divergence is a measure. In d = 1 with Ω = (−2, 2), σ ≡ 1 and f = − sign(x), one has ∇ · f = −2δ0 , purely singular; this is the field which in the whole space defeats the energy estimate of [10, Remark 3.3], and which likewise defeats the energy method above. Uniqueness nevertheless holds, by duality. The distinction is worth recording: absolute continuity of D· f is indispensable for the energy identity — and, in the whole-space theory, for the dissipation bound and for the Lagrangian hypothesis (F3) — but not for uniqueness of the density flow itself. The same phenomenon appears in [10], where drifts with singular divergence are admitted in the analysis of the density lower bound through heat-kernel rather than energy arguments. 3. Confined regular Lagrangian flow 3.1. Boundary sheet identity. The following lemma plays a central role. Lemma 3.1 (Boundary sheet identity). Let Ω be a bounded Lipschitz domain, U ⊃ ∂Ω open, and v ∈ BV (Ω ∩ U ; Rd ) ∩ BVloc (Ω; Rd ). Then the zero extension v 0 := v 1Ω belongs to BVloc (Rd ; Rd ), its interior trace tr v on ∂Ω exists in L1 (∂Ω; Hd−1 ), and (9) Dv 0 = Dv⌞Ω−(tr v)⊗ν Hd−1 ⌞∂Ω,

D·v 0 = (∇·ac v) Ld ⌞Ω+Ds·v⌞Ω−(tr v·ν) Hd−1 ⌞∂Ω.

In particular, if D · v = (∇· v)Ld ⌞Ω is absolutely continuous in Ω, then D · v 0 is absolutely continuous on Rd if and only if tr v · ν = 0 Hd−1 -a.e. on ∂Ω. Proof. The extension by zero of a BV field across a Lipschitz hypersurface is BV with the stated jump part; this is the standard trace and extension theory for BV functions on Lipschitz domains, applied on a finite cover of ∂Ω by balls B ⊂ U , with the interior estimate on compacts of Ω supplied by BVloc . Taking traces of the first identity in (9) gives the second; the equivalence follows since the two measures on the right-hand side are mutually singular. □ Thus tangency is exactly the condition removing the singular boundary part −(tr v · ν) Hd−1 ⌞∂Ω from the divergence of the zero extension: a nonzero normal trace acts as a singular compression or expansion sheet supported on ∂Ω, invisible to the absolutely continuous divergence -the boundary counterpart of the absolute-continuity requirement on the divergence in the Ambrosio–DiPerna–Lions flow theorem (Theorem 2.3). Theorem B will exhibit a vector field for which an analogous wall-localised mechanism produces an unbounded Jacobian along near-wall trajectories. 3.2. Assumptions and proof of Theorem A. Assumption 3.2 (Flux–density hypotheses (F)). Ω is a bounded Lipschitz domain, and: (F0) (p, J) is a no-flux weak solution (Definition 2.1) with p0 ∈ L1 ∩ L∞ (Ω). (F1) pt > 0 Ld -a.e. in Ω for a.e.
t, so that v := J/p is defined a.e. (F2) v ∈ L1 [0, T ]; BVloc (Ω; Rd ) , and there is an open U ⊃ ∂Ω with v ∈ L1 [0, T ]; BV (Ω ∩
U ; Rd ) .
(F3) D· vt = (∇· vt )Ld ⌞Ω for a.e. t, and [∇· v]− ∈ L1 [0, T ]; L∞ (Ω) . (F4) (Tangency) tr vt ·ν = 0 Hd−1 -a.e. on ∂Ω, for a.e. t, the trace being supplied by Lemma 3.1.

12

RAMA CONT

Here L1 ([0, T ]; BVloc ) means t 7→ ∥vt ∥BV (K) ∈ L1 (0, T ) for every compact K ⊂ Ω; global v ∈ L1t BV (Ω) implies (F2), the interior-local form being the sharp one (Section 5). No separate integrability hypothesis appears: K := Ω \ U is compact in Ω, so Z T Z T Z TZ ∥vt ∥L1 (Ω∩U ) < ∞ : ∥vt ∥L1 (K) + |v| ≤ (10) 0

Ω

0

0

global integrability is subsumed. Proof of Theorem A. Step 1: the zero extension satisfies (R1)–(R3). By Lemma 3.1 applied 0 for a.e. t, together with
a fixed finite cover of ∂Ω giving a uniform trace constant, v := v1Ω ∈ 1 d L [0, T ]; BVloc (R ) : (R1) holds. By (F3), (F4) and the equivalence in Lemma 3.1,
D· vt0 = (∇· vt )1Ω Ld , [∇· v 0 ]− = [∇· v]− 1Ω ∈ L1 [0, T ]; L∞ (Rd ) : (R2) holds. Finally v 0 is supported in the bounded set Ω̄ and lies in L1 ([0, T ] × Rd ) by (10), so (R3) holds with the L1 component alone. Theorem 2.3 yields an RLF Ẑ on Rd , unique up to null sets, with the stated compression constant. Step 2: invariance of Ω̄ (elementary). Fix x with Ẑ(0, x) = x ∈ Ω and t 7→ Ẑ(t, x) an integral curve of v 0 . The set O := {t : Ẑ(t, x) ∈ / Ω̄} is open; on any component (a, b) ⊂ O the curve lies in the open set Rd \ Ω̄, where v 0 ≡ 0, hence is constant = ξ ∈ / Ω̄ there, and by continuity Ẑ(a, x) = ξ ∈ / Ω̄. If a > 0 this places a ∈ O, contradicting that a is a left endpoint of a component; so a = 0, contradicting Ẑ(0, x) ∈ Ω. Hence O = ∅. Moreover RT d Θ− Ld (∂Ω) = 0, so a.e. trajectory spends L1 -null time on ∂Ω; 0 Ẑ(t, ·)# (L ⌞Ω)(∂Ω) dt ≤ T e 0 d since v = v L -a.e. on Ω, the integral identity holds with v in place of v 0 . Set Z := Ẑ|[0,T ]×Ω̄ . Uniqueness: any flow as in the statement, extended by the constant flow outside Ω̄, is an RLF for v 0 , hence coincides with Ẑ up to null sets. Step 3: transport. Let p̄t := pt 1Ω . For φ ∈ Cc∞ (Rd × [0, T )), using J = pv a.e. in Ω and 0 v = v on Ω, Z TZ Z Z TZ Z

0 p̄ ∂t φ + v · ∇φ + p̄0 φ(·, 0) = p ∂t φ + J · ∇φ + p0 φ(·, 0) = 0 0

Rd

Rd

0

Ω

Ω

by (NF): the no-flux formulation with unrestricted test functions is exactly the statement that the zero extension of p solves the global continuity equation — no boundary term appears because none was ever discarded. Thus p̄ is a bounded distributional solution for v 0 (with p̄v 0 = J1Ω ∈ L1 ), as is ut := Ẑ(t, ·)# (p̄0 Ld ) by Theorem 2.3(3); comparison gives ut = p̄t Ld for a.e. t, Step 2 shows ut is carried by Ω̄, and narrow continuity of both curves upgrades the identity to every t. □ 3.3. Tangency and time reversal. Corollary 3.3 (Tangency is automatic for positive continuous densities). Assume (F0)–(F3) and that for a.e. t, pt ∈ C 1 (Ω̄) with minΩ̄ pt > 0. Then (F4) holds and Theorem A applies. Proof. For ψ ∈ C 1 (Ω̄) write ψ = χψ + (1 − χ)ψ with χ ∈ Cc∞ (U ), χ ≡ 1 near ∂Ω; the (1 − χ)part is an interior identity requiring no trace. For the χ-part: for a.e. t, Jt = pt vt ∈ BV (Ω∩U ) with tr Jt = pt |∂Ω tr vt Hd−1 -a.e. Testing (NF) with φ = χψ(x)η(t), η ∈ Cc∞ (0, T ), and subtracting the interior identity ∂t p = −∇· J in D′ (Ω × (0, T )) via the Gauss–Green formula for BV fields on Ω∩U [5, 9] — only the ∂Ω portion of the boundary contributing, χψ vanishing RT R on the rest — yields 0 η ∂Ω χψ tr Jt · ν dHd−1 dt = 0 for all ψ, η; hence tr Jt · ν = 0 Hd−1 -a.e. for a.e. t, and dividing by pt |∂Ω > 0 gives (F4). □

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

13

Remark 3.4 (Sharpness of the assumptions). (a) Within the zero-extension strategy the collar BV in (F2) cannot be reduced to boundedness plus a divergence-measure condition: the Anzellotti–Chen–Frid normal trace [5, 9] would still make (F4) meaningful, but (R1) fails near ∂Ω, and renormalisation is required to operate across ∂Ω, where for merely bounded divergence-free fields uniqueness fails [14]. Sobolev regularity W 1,1 (Ω ∩ U ) serves equally well. This is a limitation of the method, however, and not of the problem: as discussed in Remark 3.5 below, the intrinsic theory on Ω developed by Crippa, De Rosa, Inversi and Nesi [13] dispenses with boundary BV regularity precisely on the exiting and tangent portions of ∂Ω, which is where the present hypotheses place us. (b) The time-integrability in the interior part of (F2) is essential: in the sharpness example of Section 5 the velocity is real-analytic RT in Ω for every t > 0, and it is precisely 0 ∥vt ∥BV (K) dt that diverges. (c) The boundary hypotheses (F2)–(F4) cannot be relaxed jointly so as to admit a boundary current: Theorem B. Whether tangency alone is indispensable is a subtler question; see Remark 4.9. Remark 3.5 (Normal traces and the intrinsic theory on Ω). Various notions of normal trace have been studied in the literature: the distributional one (Anzellotti [5], Chen–Frid [9]) for measure-divergence fields; the normal Lebesgue trace vn∂Ω introduced by De Rosa and Inversi [15], which requires the boundary value to be attained in an averaged Lebesgue sense on interior balls; and the strong BV trace of Ambrosio et al. [3, Thm. 3.87], used in Lemma 3.1. Crippa et al. [13] prove that for bounded measure-divergence fields the normal Lebesgue trace, when it exists, coincides with the distributional one and satisfies the Gauss–Green identity, and that it is a notion lying strictly between the other two. Since a BV field attains its normal Lebesgue trace, and that trace is v Ω · ν [15, Prop. 5.5], our hypothesis (F4) implies vn∂Ω ≡ 0: tangency as used here is the strongest of the three, and Lebesgue tangency is its natural weakening. Crippa et al. [13] recently established uniqueness for continuity equations on bounded domains without the global boundary BV assumption of [12] exactly on the portion of ∂Ω where characteristics exit or are tangent. In their notation, tangency forces Γ− = ∅ — no characteristic enters — so their BV -near-Γ− hypothesis is vacuous, while their uniformlyexiting condition follows from vanishing normal Lebesgue trace [13, Cor. 3.9]. Their [13, Cor. 4.6], for divergence-free fields tangent to ∂Ω in the Lebesgue sense, gives uniqueness under interior BVloc (Ω) regularity alone. This is strong evidence that the collar hypothesis in (F2) is removable for the Eulerian portion of Theorem A; what does not follow automatically is the Lagrangian result, since our flow, invariance and transport statements are all obtained from the zero extension on Rd rather than intrinsically on Ω. Two hypotheses are not comparable, though: [13] work with v ∈ L∞ and ∇ · v ∈ L1t L∞ x two-sided, whereas (F2)– (F3) allow unbounded BV fields and only a one-sided divergence bound. Conversely, the counterexample of Theorem B is unbounded near the wall and therefore lies outside the framework of [13] as well; it contradicts none of their uniqueness results. Crippa et al. [13] construct a bounded divergence-free field whose normal Lebesgue trace exists and equals −1 — characteristics entering everywhere — for which uniqueness fails. Tangency excludes exactly this configuration, which is why the present setting falls on the favourable side of their dichotomy. Remark 3.6 (Time reversal). If in addition [∇ · v]+ ∈ L1 ([0, T ]; L∞ (Ω)), the two-sided compression bound and essential invertibility of Z(t, ·) follow as in the whole space; tangency is invariant under v 7→ −v, so time reversal requires no new boundary hypothesis, and the deterministic reverse flow transports pT onto p0 under the same conditions read backwards.

14

RAMA CONT

4. The boundary current We now prove Theorem B. The construction is guided by the rigidity results of Section 6, which show (Lemma 6.4 below) that tangency is automatic wherever v is bounded near the wall. Any example in which tangency fails must therefore carry a boundary current: a tangential flux of nonvanishing linear density g(x1 , t) := J1 (x1 , 0, t) along a wall where p vanishes linearly, with (11)

v1 ≈

g(x1 , t) , c(x1 , t) x2

v2 x2 =0 = −

∂1 g(x1 , t) c(x1 , t)

(p ≈ c(x1 , t) x2 ).

4.1. Construction of the boundary current. Let Ω := T × (0, 1). Fix λ > 0, set ξ := x1 − t, m(s) := 1 + 21 cos s ∈ [ 12 , 23 ], and define J := p e1 + ∇⊥ ψ,

ψ(x, t) := πλ sin(πx2 ) cos ξ,
where ∇⊥ ψ := (∂2 ψ, −∂1 ψ) = λ cos(πx2 ) cos ξ, πλ sin(πx2 ) sin ξ . Then p ∈ C ∞ (Ω̄ × [0, T ]) R is positive in Ω, vanishes linearly on both walls, and Ω pt = 14 · 2π · π2 = 1. The continuity equation holds classically (∂t p = −∂1 p while ∇· J = ∂1 p + ∇· ∇⊥ ψ = ∂1 p), and the no-flux condition holds pointwise (J2 = πλ sin(πx2 ) sin ξ vanishes at x2 ∈ {0, 1}). The velocity is

(12)

(13)

p(x, t) := 41 sin(πx2 ) m(ξ),

v1 = 1 +

4λ cos(πx2 ) cos ξ , sin(πx2 ) m(ξ)

v2 =

4λ sin ξ . π m(ξ)

sin(x1 −t) Both components are C ∞ in Ω; v2 extends smoothly to Ω̄ with trace 4λπm(x on {x2 = 0}, 1 −t) nonzero for a.e. (x1 , t) and outward on the moving half-wall {sin(x1 − t) < 0}. The tangential cos ξ component realises the boundary current, v1 ∼ 4λ / L1 πm x2 , matching (11); consequently v ∈ near either wall and [∇·v]− is unbounded there. This proves part (ii) of Theorem B: tangency fails classically, and hypotheses (F2)–(F3) fail on the collar as well — consistently with Theorem A, which would otherwise apply and contradict part (iv). It should be emphasised that Lemma 3.1 is not available here, and that the failure of tangency cannot be expressed through it. That lemma presupposes v ∈ BV (Ω ∩ U ), hence v ∈ L1 on a collar; since v1 ∼ C/x2 , the zero extension v1Ω is not locally integrable on Rd , does not define a distribution, and has no distributional divergence — so there is no “boundary sheet” −(tr v · ν)Hd−1 ⌞∂Ω associated with this field in the sense of (5). What is available is the classical trace of v · ν, which exists because v2 happens to extend smoothly, and the following exhaustion statement, which is the precise form of the analogy. For η > 0 let Ωη := T × (η, 1 − η). On Ω̄η the field v is smooth and bounded with bounded derivatives, so Lemma 3.1 applies to Ωη and gives

D· v1Ωη = (∇· v) L2 ⌞Ωη − v · νη H1 ⌞∂Ωη , sin ξ 1 whose singular part on the lower face {x2 = η} equals 4λ πm(ξ) H , independently of η. These singular parts do not vanish as η ↓ 0; they converge weakly-∗ to the nonzero measure 4λ sin ξ 1 πm(ξ) H ⌞{x2 = 0}. The example therefore exhibits, on every interior surface parallel to the wall, exactly the singular divergence contribution that tangency excludes — while the limiting object is not itself the divergence of any extension of v.

4.2. Weighted uniqueness for no-flux Fokker–Planck equations with entrance-type singular drifts. In this section we establish a uniqueness result for no-flux Fokker–Planck equations whose drift has an entrance-type singularity at the boundary, a regime where

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

15

f ∈ / L∞ and [∇· f ]− ∈ / L∞ , so that neither the duality argument of Proposition 2.6 nor the unweighted energy method of Remark 2.7 apply. We will use this result in the sequel ( Corollary 4.5), but as it is of independent interest we formulate in more general form. Assumption 4.1 (W). Ω is a bounded domain with C 1,1 boundary, or the periodic strip T×(0, 1) (or annulus). The pair p ∈ C 1 (Ω̄×[0, T ]), J ∈ C 1 (Ω̄×[0, T ]; Rd ) satisfies, classically, ∂t p + ∇· J = 0 in Ω̄ × [0, T ],

J · ν = 0 on ∂Ω × [0, T ],

with p > 0 in Ω×[0, T ] and p(x, t) ≤ C0 dist(x, ∂Ω). Set v := J/p, σ ≡ 1, and f := v+ 12 ∇ log p on Ω, so that (p, J) solves the no-flux Fokker–Planck problem with coefficients (f, σ), by the identity f p − 21 ∇p = J. No lower bound p ≥ c dist is required. Definition 4.2 (Relative energy class). Z TZ n o
∞ Cp := q ≥ 0 : h := q/p ∈ L (0, T ) × Ω , p |∇h|2 dx dt < ∞ , 0

Ω

and q ∈ Cp is a no-flux weak solution with initial condition q0 if (4) holds with the flux Jq := f q − 12 ∇q, this being the flux of the Fokker–Planck equation written as a conservation law, ∂t q+∇·Jq = 0, and hence the field entering (4). Two remarks make this meaningful. First, in terms of h the flux is
(14) Jq = v + 12 ∇ log p hp − 12 ∇(hp) = h J − 12 p ∇h, consistently with Jp R= J for h ≡ 1, Rin which every singular object is absorbed: p∇h ∈ L2 ((0, T ) × Ω) since |p∇h|2 ≤ ∥p∥∞ p|∇h|2 , and hJ ∈ L∞ ; all terms of the weak formulation are finite even though f ∈ / L2 (Ω). Second, p ∈ Cp (with h ≡ 1) and is a no-flux weak solution with initial condition p0 , by (W) and the classical divergence theorem. Theorem 4.3 (Weighted uniqueness). Under Assumption (W), for every initial condition there is at most one no-flux weak solution in Cp . In particular, (pt ) is the unique no-flux weak solution in Cp with initial condition p0 . The proof rests on the observation that testing the equation for the ratio h = q/p against h in the weighted space L2 (p dx) produces transport terms that cancel exactly against the time derivative of the weight, because p solves the same continuity equation. The singular drift never appears, and neither does its divergence. Proof. Let q 1 , q 2 ∈ Cp solve with the same initial condition; set u := q 1 − q 2 , w := u/p, M := ∥w∥∞ . Subtracting the weak formulations and using (14), Z TZ Z TZ   1 p∇w − wJ · ∇φ ∀ φ ∈ C ∞ (Ω̄ × [0, T )). (15) p w ∂t φ = 2 0

Ω

0

Ω

Step 1: localisation. Write ρ(x) := dist(x, ∂Ω), which is C 1,1 in a collar {ρ < ρ0 } with ∇ρ = −ν on ∂Ω. For δ ∈ (0, ρ0 /4) pick χδ := θδ ◦ ρ with θδ ∈ C ∞ ([0, ∞); [0, 1]), θδ = 0 on [0, δ], θδ = 1 on [2δ, ∞), |θδ′ | ≤ Cδ −1 (on the strip take χδ = χδ (x2 ) directly). Set Ωδ := {ρ > δ}, on whose closure p is bounded above and below by positive constants uniformly in t. Then q i ∈ Cp gives u ∈ L2 ((0, T ); H 1 (Ωδ )), and by (15) — whose right-hand side is a

16

RAMA CONT

bounded functional of ∇φ ∈ L2 — also ∂t u ∈ L2 ((0, T ); H −1 (Ωδ )). By density (both sides of (15) being continuous in the relevant norms, using p∇w ∈ L2 and wJ ∈ L∞ ), (15) extends to test functions in L2 ((0, T ); H01 (Ωδ )) with square-integrable time derivative vanishing at T ; and the standard theory (Lions–Magenes duality [35]) yields u ∈ C([0, T ]; L2 (Ωδ )) with u(0) = 0, the initial data being equal. Step 2: weighted energy identity. Let ζ := χδ /p ∈ C 1 (Ω̄δ × [0, T ]), bounded with bounded time derivative there. The chain rule for ⟨∂t u, ζu⟩ (legitimate since ζu = wχδ ∈ L2 ((0, T ); H01 (Ωδ ))) gives, for a.e. t, Z Z d 2 ∂t p w2 χδ . p w χδ = 2 ∂t u, wχδ − dt Ω Ω Substituting (15) with test function wχδ and expanding ∇(wχδ ) = χδ ∇w + w∇χδ , Z Z Z Z 2 ∂t u, wχδ = − p|∇w|2 χδ +2 wJ·∇w χδ +Eδ (t), Eδ := − p(∇w·∇χδ )w+2 w2 J·∇χδ . Ω

Ω

Ω

Ω

Step 3: exact cancellation. On supp χδ the function w lies in H 1 ∩ L∞ , so w2 ∈ W 1,1 with ∇(w2 ) = 2w∇w; J is C 1 ; integrating by parts against χδ ∈ Cc∞ and using ∂t p = −∇ · J classically, Z Z Z Z
2 2 ∂ t p w χδ = ∇· J w χδ = − (J · ∇χδ ) w2 . 2 wJ · ∇w χδ − Ω

Ω

Ω

Ω

This is the heart of the proof: the transport term produced by the flux and the term produced by differentiating the weight combine into a pure divergence, which sees only the R component J ·∇χδ ∝ J · ∇ρ on the cutoff layer. The individually divergent quantities (note |J|2 /p = ∞ in the example of Section 4) are never separated. Collecting the terms we obtain Z Z Z d 2 2 e e (16) p w χδ = − p|∇w| χδ + Eδ (t), Eδ := Eδ − (J · ∇χδ ) w2 . dt Ω Ω Ω Step 4: the layer errors vanish. Let Lδ := {δ < ρ < 2δ} ⊃ supp ∇χδ , of measure ≤ Cδ. Three facts are used: p ≤ 2C0 δ on Lδ (by assumption); |J · ∇ρ| ≤ C1 ρ ≤ 2C1 δ on Lδ , which follows from J ∈ C 1 (Ω̄), ρ ∈ C 1,1 , and J · ∇ρ = −J · ν = 0 on ∂Ω — the no-flux condition doing its work; and, decisively, that ∇χδ is parallel to ∇ρ, so the cutoff never meets the tangential component of J, the carrier of any boundary current. Then, integrating in time, Z TZ C w2 (J · ∇χδ ) ≤ M 2 · 2C1 δ · T |Lδ | ≤ C ′ M 2 T δ −−→ 0, δ↓0 δ 0 and by Cauchy–Schwarz with weight p, Z TZ Z Z 1/2  1/2 √ CM  T p(∇w · ∇χδ )w ≤ p|∇w|2 T |Lδ | sup p ≤ C ′′ M T εδ , δ Lδ 0 0 Lδ RTR where εδ := ( 0 Lδ p|∇w|2 )1/2 → 0 as the tail of a convergent integral. The scaling deserves R emphasis: the factor δ −1 from the cutoff is beaten by ( Lδ p)1/2 ≲ δ, which is precisely where p ≲ dist is used: the weight vanishes fast enough at the wall that the boundary is invisible to the energy. Discarding the negative dissipation in (16) and integrating from 0 to t with u(0) = 0, Z √
pt wt2 χδ ≤ C M T εδ + M 2 T δ for a.e. t. Ω

Letting δ ↓ 0: monotone convergence on the left, zero on the right; hence wt = 0 pt -a.e., and since pt > 0 in Ω, q 1 = q 2 a.e. □

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

17

Remark 4.4 (Interpretation). From a probabilistic perspective, h = q/pR is a Doob htransform ratio, and the identity of Steps 2–3 isRthe decay of the χ2 -divergence (q/p−1)2 p dx with dissipation the relative Dirichlet energy p|∇(q/p)|2 ; the class Cp is the natural relative analogue of the energy class XΩ of Proposition 2.6. A variant of the same computation applied to (w − k)2+ yields the comparison principle q0 ≤ kp0 ⇒ qt ≤ kpt . The domination q ≤ Cp forbids competing solutions from carrying mass toward the walls faster than p does; uniqueness in the unweighted class remains open (Section 7). 4.3. Eulerian perspective: uniqueness of the density evolution. Setting σ ≡ 1 and f := v + 21 ∇ log p, the pair (p, J) satisfies, identically, the no-flux Fokker–Planck problem with coefficients (f, σ), by the identity f p − 12 ∇p = J. Near the walls, f has the Bessel-type entrance singularity f2 ≈ 2x12 (the associated diffusion never reaches the wall and no local time appears), and f ∈ / L∞ (Ω × (0, T )), while [∇· f ]− ≍ dist(x, ∂Ω)−2 is unbounded: neither Proposition 2.6 nor its energy counterpart of Remark 2.7 is applicable. The Assumption (W) of Section 4.2 holds by inspection (p ≤ 3π 8 dist(x, ∂Ω)), whence: Corollary 4.5. The density flow (pt ) of (12) is the unique no-flux weak solution for the Fokker–Planck equation with initial condition p0 , in the relative energy class Cp of Theorem 4.3. This proves part (i) of Theorem B. 4.4. Characteristics, marginal transport, and failure of compressibility. We now give the proof of (iii) and (iv). We isolate the quantitative mechanism as a lemma. Let m(ξ) = 1 + 21 cos ξ ∈ [ 12 , 23 ], and (17)

cλ :=

4λ , 3π

Cλ :=

8λ , π

t̄ :=

1 π = . 8Cλ 64λ

sin ξ 3 1 From v2 = 4λ πm(ξ) and 2 ≤ m ≤ 2 one has the two-sided control

(18)

|v2 | ≤ Cλ everywhere,

v2 ≥ cλ wherever sin ξ ≥ 21 .

The upper bound confines the trajectory and the lower bound makes it climb. 1 Lemma 4.6 (Transit lemma). Let t∗ ∈ (0, t̄ ] and, for 0 < κ < 100 , put o n
Bκ := (ξ, x2 ) : ξ ∈ π6 , π3 , κ < ψ̄(ξ, x2 ) < 2κ, x2 < 12 ,

a nonempty open subset of Ω in the co-moving frame. Then: (a) p ≤ 32 κ on Bκ ; (b) every frame trajectory starting in Bκ remains in {x2 < 12 } on [0, t∗ ], with sin ξ ≥ 12 throughout, and satisfies
x2 (t∗ ) ≥ cλ t∗1; (c) consequently p Z(t∗ , z) ≥ c(t∗ ) := 8 sin(πcλ t∗ ) > 0 for every z ∈ Bκ , with c(t∗ ) independent of κ; (d) Aκ := Z(t∗ , Bκ ) is open with L2 (Aκ ) > 0. √

1 1 Proof. (a) On Bκ , cos ξ ∈ ( 12 , 23 ), so sin(πx2 ) = ψ̄/ cos ξ < 2κ/ 21 = 4κ < 25 ; hence x2 < 50 1 1 3 3 and p = 4 sin(πx2 )m(ξ) ≤ 4 · 4κ · 2 = 2 κ. (b) On the orbit through such a point, cos ξ = ψ̄/ sin(πx2 ) > 0, so ξ ∈ (− π2 , π2 ) for 2 ) cos ξ all time. As long as x2 < 12 we have ξ˙ = 4λ cos(πx > 0, so ξ increases and remains sin(πx2 )m π π 1 in [ 6 , 2 ); hence sin ξ ≥ 2 and, by (18), cλ ≤ ẋ2 ≤ Cλ . Integrating the upper bound,

18

RAMA CONT

1 1 x2 (t) ≤ x2 (0) + Cλ t ≤ 50 + Cλ t̄ = 50 + 18 < 21 for t ≤ t∗ , so the constraint x2 < 12 is never violated and the bootstrap closes by continuity. Integrating the lower bound gives x2 (t∗ ) ≥ cλ t∗ . (c) By (b), x2 (t∗ ) ∈ [cλ t∗ , 12 ], so πx2 (t∗ ) ∈ [πcλ t∗ , π2 ] and, sin being increasing there, sin(πx2 (t∗ )) ≥ sin(πcλ t∗ ); with m ≥ 12 , p ≥ 14 sin(πcλ t∗ ) · 12 = c(t∗ ). (d) By (b) the trajectories issuing from Bκ stay in the open region {0 < x2 < 21 , cos ξ > 0} on [0, t∗ ], where the frame field v fr = ∇⊥ ψ/p is C ∞ . Hence Z(t∗ , ·) is a C ∞ diffeomorphism of a neighbourhood of B̄κ onto its image, by smooth dependence on initial conditions and invertibility of the backward flow; the image of the nonempty open set Bκ is therefore open and of positive measure. □

Proof of Theorem B, parts (iii)–(iv). (iii). In the co-moving frame (ξ, x2 ) the ODE Ż = v(Z, t) is autonomous: ξ˙ = v1 − 1, ẋ2 = v2 define the field v fr = ∇⊥ ψ/p, with ψ, p read d as functions of (ξ, x2 ). Then dt ψ(ξ(t), x2 (t)) = ∇ψ · v fr = 0: ψ̄ := sin(πx2 ) cos ξ is a first integral, and orbits lie on its level sets. The level set {ψ̄ = 0} is the union of the two walls and the two vertical circles {cos ξ = 0}, a Lebesgue-null set; every other level {ψ̄ = κ}, 0 < |κ| < 1, is a closed curve in Ω encircling (ξ, x2 ) = (0, 21 ) (for κ > 0; symmetrically for κ < 0), with a lower arc x2 = π1 arcsin(κ/ cos ξ) hugging the bottom wall, an upper arc hugging the top wall, and turning points at cos ξ = κ, x2 = 12 . On these curves v is smooth and nonvanishing: solutions are unique, periodic, and confined. The frame field satisfies ∇· (p v fr ) = ∇· ∇⊥ ψ = 0, so the frame flow preserves the measure p dξ dx2 ; translating to the lab frame gives Z(t, ·)# (p0 L2 ) = pt L2 . (iv). Let Z satisfy the trajectory condition of Definition 2.2(i). Off the null set {ψ̄ = 0} the field is smooth, so Z agrees a.e. with the flow of (iii); it suffices to disprove the compressibility bound for that flow. Since the frame flow preserves p L2 , the push-forward Z(t, ·)# L2 has density p(y)
. ρt (y) = p Z(t, ·)−1 (y) 1 Fix T > 0 and set t∗ := min{T, t̄ } ∈ (0, T ] with t̄ as in (17). Let κ ∈ (0, 100 ) and let Bκ , −1 Aκ = Z(t∗ , Bκ ) be as in Lemma 4.6. For y ∈ Aκ write z := Z(t∗ , ·) (y) ∈ Bκ ; then by parts (a) and (c) of that lemma,

ρt∗ (y) =

c(t∗ ) p(y) ≥ 3 −−→ ∞, κ↓0 p(z) 2κ

while L2 (Aκ ) > 0 by part (d). Hence for every constant L there is κ with Z(t∗ , ·)# L2 ≥ LL2 on a set of positive measure. Since Definition 2.2(ii) demands a single constant L valid for all t ∈ [0, T ], and this fails at t = t∗ ∈ (0, T ], no regular Lagrangian flow exists on [0, T ]. □ Remark 4.7. The orbits of (iii) are periodic in the co-moving frame, so a trajectory started near the wall passes through the interior and may return near the wall; nothing in the construction guarantees that the compression witnessed above is still present at the exact terminal time T , and for large T a statement about Z(T, ·)# L2 alone would require an analysis of the period map. None is needed: failure of the compressibility bound at a single time t∗ ∈ (0, T ] already contradicts Definition 2.2(ii), which quantifies over all t ∈ [0, T ]. The numerical experiment of Section 4.5 measures supt≤T ρt , which is precisely the quantity appearing in the corrected statement.

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

19

Remark 4.8 (A wall-localised compression mechanism). The infinite compression in part (iv) is a wall-localised infinite-compression mechanism analogous to the singular boundary divergence sheet that Theorem A excludes through tangency; the two are not identified by Lemma 3.1, which does not apply to this field, but by the exhaustion computation above: on each surface {x2 = η} the extension of v carries a nonzero singular divergence of size independent of η, and it is along the orbits threading those surfaces that the Jacobian degenerates. The example separates three layers that the whole-space theory tends to fuse: existence of invariant characteristics (true here, by the first integral, and guaranteed in general by Proposition 6.2); pointwise deterministic transport of the marginals (true here); and the regular Lagrangian flow property (false here, for every T > 0). Remark 4.9. Since v ∈ / L1 on every collar, the example violates (F2) and (F3) as well as (F4). It therefore does not show that tangency alone is indispensable in Theorem A; what it shows is that the collection of boundary hypotheses (F2)–(F4) cannot be relaxed far enough to admit the boundary-current mechanism. This is not an accident of the construction. In the regime of Lemma 6.4 the hypotheses are structurally entangled: by Corollary 6.5 below, whenever the density vanishes linearly at a C 1 wall and the flux is C 1 up to it, failure of tangency forces v ∈ / L1 near that wall. Within this class, therefore, no example can isolate (F4), and it is unclear whether tangency is independently necessary. 4.5. Numerical illustration. Figure 1 displays the mechanism, for λ = 1 and T = 0.5. Panel (a) shows the orbit structure in the co-moving frame: level sets of the first integral ψ̄ = sin(πx2 ) cos ξ, closed curves encircling the elliptic centres and hugging both walls, with the separatrix {ψ̄ = 0} (walls and the circles {cos ξ = 0}) dashed. Panel (b) is the trace of the normal velocity on the bottom wall, v2 (ξ, 0) = 4λ sin ξ/(πm(ξ)): nonzero for a.e. ξ and outward on the moving half-wall, the failure of tangency in Theorem B(ii) made visible. Panels (c) and (d) quantify part (iv). For each κ, a trajectory is started at the bottom of the lower arc of the orbit {ψ̄ = κ} and integrated with an adaptive high-order scheme; the drift supt≤T |ψ̄(Zt ) − κ| remains below 10−11 throughout. The witnessed compression, measured as supt≤T ρt in accordance with Theorem B(iv) and Remark 4.7, supt≤T ρt , with ρt = p(Zt )/p(Z0 ), follows the predicted law c(T )/κ over two decades: 0.1 0.0316 0.01 0.0032 0.001 κ supt≤T ρt 7.00 21.4 67.0 211 667 κ · supt≤T ρt 0.700 0.677 0.670 0.668 0.667 so that c(T ) ≈ 23 at T = 0.5: the “elementary phase-plane estimates” invoked in the proof of Theorem B(iv) are confirmed with the constant included. Panel (d) shows the density ρT (y) = p(y)/p(Z(T, ·)−1 (y)) of the push-forward Z(T, ·)# L2 , computed pointwise from the exact formula by backward integration on a grid: the compression (red) concentrates in bands along the walls downstream of the turning regions, the depletion (blue) in the regions the near-wall mass has vacated — the wall-localised compression mechanism described in Remark 4.8. A numerical illustration of the data-endpoint example of Section 5.2 would essentially reproduce the corresponding figure of [10], the image corrections being exponentially small at plotting scale, and is omitted. 5. Mimicking reflected diffusions with differential equations 5.1. The reflection term disappears from the probability flow. Consider the Skorokhod problem in a bounded C 1,1 domain, (19)

dXt = f (Xt , t) dt + σ(t) dWt − ν(Xt ) dLt ,

Xt ∈ Ω̄,

X0 ∼ µ0 ∈ P(Ω̄),

20

RAMA CONT

(a) orbits: level sets of = sin( x2)cos

1.0

(b) trace of the normal velocity on {x2 = 0} 1.5 1.0

0.8

0.5

0.4

v2( , 0)

x2

{cos = 0}

0.6

0.5

0.2

wall 3

2

1

0

= x1 t

1.5 1

2

3

3

(c) T c(T)/ , T = 0.5

2

1

0

1

2

(d) density of Z(T, )# 2 (exact formula)

1.0

sup t along orbit = t T slope 1: c(T)/

0.8 0.6

102

0.4 0.2 101 10 3

10 2

10 1

0.0

3

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

x2

compression ratio

outward (tangency fails)

1.0

log10 T

0.0

0.0

3

2

1

0

1

2

3

Figure 1. The boundary-current example (12), λ = 1. (a) Orbits of the probability-flow ODE in the co-moving frame: level sets of ψ̄; the two highlighted orbits (κ = 0.04, 0.12) hug both walls and pass through mid-height at the turns; separatrix dashed; arrows indicate the direction of motion. (b) Trace of the normal velocity on the bottom wall: nonzero a.e., outward on half the wall (shaded) — tangency fails, yet the normal flux p v2 vanishes because p does. (c) Witnessed compression along the orbit {ψ̄ = κ} up to time T = 0.5, against the predicted c(T )/κ (slope −1): no uniform compressibility constant exists. (d) Push-forward density ρT of Lebesgue measure under the flow, from the exact formula ρT (y) = p(y)/p(Z(T, ·)−1 (y)): compression concentrates along the walls downstream of the turning regions.

with σ scalar, 0 < ϵ ≤ σ ≤ ϵ−1 , and L the boundary local time. For bounded measurable f , (19) is well posed in law — by the Skorokhod-map theory for normal reflection in smooth domains [27], extended to far more general reflection fields and domains by the convexduality and extended-Skorokhod-problem framework [17, 31] and the submartingale-problem characterisation [24]; for t > 0 the marginal law has a density pt , Hölder continuous and strictly positive on Ω̄ and solves the Fokker–Planck equation with co-normal no-flux condition ( 21 σ 2 ∇pt − f pt ) · ν = 0 on ∂Ω. Regularity beyond Hölder continuity requires regularity of the coefficients and is quantified in Corollary 5.1 below. Define the probability-flow velocity and flux (20)

v := f − 12 σ 2 ∇ log pt ,

J := f pt − 21 σ 2 ∇pt = pt v.

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

21

The interior equation is the continuity equation ∂t p+∇·(pv) = 0, and the boundary condition reads exactly (21)

pt v · ν = 0 on ∂Ω,

hence

v · ν = 0 on ∂Ω for t > 0,

whenever pt is positive and differentiable up to the boundary: the normal component of the score exactly cancels f · ν, and the boundary local time — a genuinely stochastic object — is entirely absorbed into ∇ log pt . The probability-flow ODE therefore carries no local time term, in contrast with the reverse-time SDE of a reflected diffusion, which retains one [7]. This is what makes deterministic samplers for constrained diffusion models [29] conceivable. The pair (20) is a no-flux weak solution in the sense of Definition 2.1, and the question whether the density flow (pt ) may be ’mimicked’ by a deterministic confined transport is answered by Theorem A: Corollary 5.1 (Early-stopped reflected transport). Let ∂Ω ∈ C 2,α , let σ satisfy 0 < ϵ ≤ σ ≤ ϵ−1 with σ 2 ∈ C α/2 ([0, T ]), and let the drift satisfy
(22) f ∈ C 1+α,(1+α)/2 Ω̄ × [0, T ]; Rd , so that in particular ∇· f ∈ C α,α/2 (Ω̄ × [0, T ]) and supt ∥f (·, t)∥C 1 (Ω̄) < ∞. No compatibility condition on the initial law is imposed: the estimates below are interior in time. Let X solve (19) with arbitrary initial law µ0 ∈ P(Ω̄), possibly singular. Then for every δ ∈ (0, T ) the pair (20) satisfies hypotheses (F0)–(F4) on [δ, T ], and the probability-flow ODE generates a regular Lagrangian flow on Ω̄ transporting pδ onto pt for every t ∈ [δ, T ], with no reflection mechanism; in fact [∇· v]± ∈ L∞ , so the two-sided bound of Remark 3.6 holds as well. The constants degenerate as δ ↓ 0. Proof. Step 1 (regularity and positivity). Under (22), σ 2 ∈ C α/2 and ∂Ω ∈ C 2,α , all coefficients of the co-normal (oblique) problem are parabolically Hölder continuous. Fix δ ∈ (0, T ). On Ω̄ × [δ/2, T ] the density is a bounded weak solution — by De Giorgi–Nash–Moser theory and the Gaussian bounds recalled above — and boundary Schauder estimates for the co-normal problem [25] are interior in time: they bound ∥p∥C 2+α,1+α/2 (Ω̄×[δ,T ]) in terms of ∥p∥L∞ (Ω̄×[δ/2,T ]) , the coefficient norms, and (δ/2)−1 , with no compatibility condition at the left endpoint. Restricting to [δ, T ] gives p ∈ C 2+α,1+α/2 (Ω̄×[δ, T ]). This route avoids treating δ as an initial parabolic time, and hence avoids assuming the very boundary regularity of pδ that is being established. Moreover p > 0 on Ω̄ × [δ, T ]: interior positivity follows from the Harnack inequality, and a zero at a boundary point (x0 , t0 ) would force ∂ν p(x0 , t0 ) < 0 by the Hopf lemma, contradicting the co-normal condition 12 σ 2 ∂ν p = (f · ν) p = 0 at that point. Set cδ := inf Ω̄×[δ,T ] p > 0. Step 2 (F0)–(F1). Re-basing at time δ, the pair (pt , Jt )t∈[δ,T ] with initial condition pδ ∈ 2,α C (Ω̄) ⊂ L1 ∩ L∞ (Ω) is a classical solution of the no-flux problem, hence a no-flux weak solution in the sense of Definition 2.1 by the divergence theorem; t 7→ pt Ld is narrowly continuous, and p ≥ cδ > 0 gives (F1). Step 3 (F2). By Step 1 and p ≥ cδ , ∇ log p = ∇p/p ∈ C 1,α (Ω̄; Rd ) with norms uniform on [δ, T ]; by (22), f (·, t) ∈ C 1 (Ω̄) ⊂ W 1,∞ (Ω) with uniformly bounded norms. Hence

v = f − 21 σ 2 ∇ log p ∈ L∞ [δ, T ]; W 1,∞ (Ω; Rd ) ⊂ L1 [δ, T ]; BV (Ω; Rd ) , measurability in t following from that of f and σ and continuity of t 7→ pt in C 2 (Ω̄). Global BV on Ω implies both the interior-local and the collar
parts of (F2). Step 4 (F3). For a.e. t, D·vt = ∇·ft − 21 σ 2 ∆ log pt Ld is absolutely continuous: ft ∈ C 1 (Ω̄) makes ∇· ft a bounded continuous function, and ∆ log p = ∆p/p − |∇p|2 /p2 ∈ C α (Ω̄) with

22

RAMA CONT

bounds uniform on [δ, T ] by Step 1. Hence [∇· v]± ∈ L∞ ([δ, T ] × Ω), which is (F3) with a two-sided bonus. Step 5 (F4). Dividing the co-normal condition by p > 0 gives, pointwise on ∂Ω, v · ν = f · ν − 12 σ 2 ∂ν log p = 0; for a field in C(Ω̄) ∩ W 1,∞ (Ω) the classical boundary values coincide with the interior BV trace, so (F4) holds. Step 6. Theorem A, applied on [δ, T ] with initial condition pδ , yields the regular Lagrangian flow and the transport identity; the constants involve c−1 δ and the Schauder norms on [δ, T ], which degenerate as δ ↓ 0. □ Remark 5.2 (The role of the drift assumptions). In nondivergence form the equation reads ∂t p − 12 σ 2 ∆p + f · ∇p + (∇· f ) p = 0, whose zeroth-order coefficient is ∇ · f . Schauder theory requires this coefficient, and not merely f , to be parabolically Hölder continuous, which is why (22) imposes f ∈ C 1+α,(1+α)/2 rather than f ∈ C α,α/2 . The latter gives ∇ · f bounded but not Hölder, and yields only first-derivative Hölder regularity of p, insufficient for the Hessian control needed in Step 4. Corollary 5.1 may also be stated conditionally: whenever the reflected problem is well posed and the probability-flow velocity satisfies (F2)–(F3) on [δ, T ], Theorem A applies, tangency being supplied by Corollary 3.3 as soon as p is C 1 and positive up to the boundary; the drift class (22) is one verifiable instance, and the rougher set below a second. In the regularity class of Corollary 5.1 the field v is spatially Lipschitz on [δ, T ] and the flow is in fact classical; the content of the corollary is that the initial law may be arbitrary — early stopping regularises the score, not the drift. The drift hypotheses cannot be dispensed with by regularity of p alone: hypotheses (F2)–(F3) constrain v = f − 12 σ 2 ∇ log p jointly, and for merely bounded measurable f neither BV regularity nor an absolutely continuous, onesidedly bounded divergence of v is available, however smooth the score part may be. This mirrors the whole-space Lagrangian theorem [10, Theorem 4.5], which likewise imposes joint drift–score hypotheses. Beyond the Schauder class (22), in which v is Lipschitz, the natural rougher hypotheses are the bounded-domain transcription of (DL ) + (S) of [10]: for the drift,

f ∈ L1 [0, T ]; BV (Ω; Rd ) , D· ft absolutely continuous, [∇· f ]− ∈ L1 [0, T ]; L∞ (Ω) ,
and for the score, ∇ log p ∈ L1 [δ, T ]; BV (Ω; Rd ) with [∆ log p]+ ∈ L1 ([δ, T ]; L∞ (Ω)), together with tangency. These give (F2)–(F4) directly and are where Theorem A earns its generality; unlike the Schauder class, however, the score conditions are then hypotheses rather than consequences, since parabolic regularity theory does not reach them from rough coefficients. Remark 5.3 (Bounded measurable drift: superposition, not flow). For merely bounded measurable f , the marginal density is Hölder continuous and strictly positive on Ω̄ × [δ, T ], as recalled above, but need not be C 1 , and no flow-level conclusion is claimed. What survives is a characteristics-level statement, and it must be rebased at a positive time: for a singular initial law the flux need not be square integrable up to t = 0. Fix δ ∈ (0, T ). On [δ, T ] the density is bounded, bounded away from 0, and satisfies the parabolic energy estimate, so J = f p − 12 σ 2 ∇p ∈ L2 ([δ, T ] × Ω) ⊂ L1 ([δ, T ] × Ω), and p > 0 there makes the vacuum hypothesis (25) vacuous. Proposition 6.2, applied to the evolution rebased at pδ , therefore yields a superposition of integral curves of v confined to Ω̄ on [δ, T ] with marginals pt . Whether a regular Lagrangian flow exists in this generality, and what happens on (0, δ), are open. The following corollary treats the case of two widely used diffusion models [33].

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

23

Corollary 5.4 (Reflected variance-exploding and variance-preserving models). Let ∂Ω ∈ C 2,α and let X solve (19) with either (VE) f ≡ 0 and 0 < ϵ ≤ σ(t) ≤ ϵ−1 measurable, or p (VP) f (x, t) = −β(t)x and σ(t) = 2β(t) with 0 < βmin ≤ β(t) ≤ βmax and β ∈ C (1+α)/2 ([0, T ]) — satisfied by the schedules used in practice, which are smooth, the reflected analogues of the forward diffusions used in score-based generative models. Then for arbitrary, possibly singular, initial laws µ0 ∈ P(Ω̄) and every δ ∈ (0, T ), the probabilityflow velocity satisfies (F0)–(F4) on [δ, T ] and the probability-flow ODE generates a regular Lagrangian flow on Ω̄ transporting pδ onto pt , t ∈ [δ, T ]. Proof. (VP). The drift is spatially affine, so ∇f = −β(t)I and ∇· f = −d β(t) are as regular 1+α,(1+α)/2 (Ω̄ × in time as β and constant in space; with β ∈ C (1+α)/2 ([0, T ]) one has
f ∈C d [0, T ]; R ), which is (22), and supt ∥f (·, t)∥C 1 (Ω̄) ≤ βmax 1 + supΩ̄ |x| < ∞ — boundedness of the domain doing what linear-growth conditions do in the whole space. Moreover σ 2 = 2β ∈ C α/2 ([0, T ]) with 2βmin ≤ σ 2 ≤ 2βmax ; the time-Hölder hypothesis on β is the analogue of the time-regularity imposed in [10] for parabolic regularity of the score. The affine structure is what makes this case safe: ∇· f is a function of t alone, so the zeroth-order coefficient of the nondivergence form is automatically Hölder. All hypotheses of Corollary 5.1 hold, and its conclusion is the claim. Note that f · ν = −β x · ν ̸≡ 0, so the boundary condition is genuinely co-normal (Remark 5.5) and Step 5 of the proof of Corollary 5.1 uses it in full. (VE). Here σ is merely measurable and the Schauder input of Corollary 5.1 is not directly Rt available; a deterministic time change supplies it. Let τ (t) := 0 12 σ(s)2 ds, a bi-Lipschitz increasing bijection of [0, T ] onto [0, τ (T )] with 12 ϵ2 ≤ τ ′ ≤ 12 ϵ−2 . Then p̃τ := pt(τ ) solves the Neumann heat equation ∂τ p̃ = ∆p̃, and Step 1 of the proof
of Corollary 5.1 (with f ≡ 0 and unit diffusion) gives p̃ ∈ C 2+α,1+α/2 Ω̄ × [τ (δ), τ (T )] with inf p̃ > 0. Back in the original time variable, supt∈[δ,T ] ∥pt ∥C 2,α (Ω̄) < ∞, inf Ω̄×[δ,T ] p > 0, and t 7→ pt is continuous into C 2 (Ω̄). Steps 2–5 of that proof now apply verbatim in the original time variable: v = − 21 σ(t)2 ∇ log pt ∈ L∞ ([δ, T ]; W 1,∞ (Ω; Rd )), the bounded measurable factor σ 2 being harmless; D·vt = − 21 σ 2 ∆ log pt Ld is absolutely continuous with a two-sided L∞ bound; and the Neumann condition ∂ν p = 0 gives v · ν = 0 pointwise. Theorem A concludes. □ Remark 5.5 (No-flux versus Neumann; loss of Gaussian structure). Two features distinguish the reflected models from their whole-space counterparts. First, for (VE) the no-flux condition reduces to the homogeneous Neumann condition ∂ν p = 0, but for (VP) it is genuinely co-normal: ∂ν p = (2f · ν/σ 2 ) p with f · ν = −β x · ν ̸≡ 0, a drift-dependent Robin-type relation; imposing plain Neumann instead would violate conservation of mass. Second, reflection destroys the variation-of-constants representation: the marginals of reflected Ornstein– Uhlenbeck dynamics are not Gaussian convolutions of the initial law, so the explicit score bounds available in the whole space have no direct analogue, and the regularity input for Corollary 5.4 comes instead from Neumann parabolic theory [25]. The example of Section 5.2 is exactly the reflected (VE) model with σ ≡ 1 started from compactly supported data, and shows the early stopping in Corollary 5.4 is necessary. Remark 5.6 (Relation to the Skorokhod-problem and probabilistic literature). The wellposedness statement above and the formulation (NF) relate to a substantial probabilistic literature on reflected diffusions. First, the Skorokhod-map theory [6, 27, 32, 17, 31] achieves pathwise stability of the reflected SDE through the constraining mechanism, whereas Theorem A achieves flow-level well-posedness of the ODE with the constraining mechanism absorbed

24

RAMA CONT

entirely into the score: two opposite resolutions of the same confinement problem. Second, (NF) is the time-dependent, normal-reflection, absolutely continuous case of the basic adjoint relationship used to characterise stationary distributions of reflected diffusions [21, 23], which in general involves a pair of measures — an interior measure and a boundary measure, the Eulerian shadow of the local time. For normal reflection with nondegenerate diffusion and a density positive up to the boundary, the boundary measure is determined by the trace of p and the pair collapses to a single density with the co-normal condition: the Eulerian counterpart of the observation (21) that the local time disappears from the probability-flow velocity. For oblique reflection the tangential component of the reflection field generically induces a surface flux along ∂Ω, and in the piecewise-smooth and degenerate settings of [24, 23] the marginals may charge the boundary, so the measure-pair formulation is unavoidable — a singular cousin of the boundary current of Theorem B. Finally, in nonsmooth domains the constraining term may fail to have bounded variation and the reflected process is only a Dirichlet process [22]: the pathwise counterpart of the degradation of score regularity at non-convex boundaries discussed in Remark 5.7 below. Remark 5.7 (Geometry influences the score). In the classical regime more is true: for reflected Brownian motion or reflected Ornstein–Uhlenbeck dynamics in a bounded convex C 1,1 domain, with 0 < c0 ≤ p0 ≤ C0 , log p0 ∈ C 1,1 (Ω̄) and the Neumann compatibility ∇ log p0 · ν = 2f · ν/σ 2 on ∂Ω, log-Hessian bounds propagate along the Neumann semigroup — convexity making the boundary contribution to the Bakry–Émery curvature nonnegative [36] — and the score is Lipschitz up to Ω̄ uniformly on [0, T ]. Convexity is not decorative: at a re-entrant corner, second-derivative estimates for the Neumann problem degrade and ∇2 log pt may blow up at the boundary even for smooth interior data. The geometry of ∂Ω thus enters the regularity theory of the score, a phenomenon absent from the whole-space setting [10]. 5.2. Failure at the endpoint: the boundary is innocent. The early-stopping in Corollary 5.1 is necessary: we now show that on the full interval [0, T ], Eulerian well-posedness does not imply deterministic transport, through a mechanism inherited from the data — and that the reflecting boundary is provably innocent. Consider reflected Brownian motion in Ω̄ = [−2, 2]: (23)

d = 1,

f ≡ 0,

σ ≡ 1,

p0 = 21 1[−1,1] .

With f ≡ 0, Proposition 2.6 applies trivially and gives uniqueness in XΩ outright. The marginal flow is given by the method of images: with φt the heat kernel, qt := p0 ∗ φt the whole-line solution, and the 8-periodic even extension of p0 across x = ±2, i Xh (24) pt (x) = qt (x − 8n) + qt (4 − x − 8n) , x ∈ [−2, 2], t > 0, n∈Z

smooth, strictly positive on [−2, 2] for t > 0, with ∂x pt (±2) = 0 and ∥∂x pt ∥2L2 = O(t−1/2 ), hence in XΩ : it is the no-flux density evolution for (23). Note v(±2, t) = − 21 ∂x log pt (±2) = 0 for every t > 0: tangency holds at the walls at all positive times. Lemma 5.8 (Edge asymptotics). Fix a compact K ⊂ (1, 2). For x ∈ K, as t ↓ 0, (x − 1)2 − log(x − 1) + 12 log t + c + O(t), 2t uniformly on K, and symmetrically on compacts of (−2, −1). log pt (x) = −

v(x, t) =

x−1 + O(1), 2t

NO-FLUX CONTINUITY EQUATIONS AND CONFINED LAGRANGIAN FLOWS

25

2 

 x+1 e−u √ √ √ Proof. For x > 1, qt (x) = 41 erfc x−1 − erfc ; inserting erfc(u) = 1 − 2u1 2 + u π 2t 2t
√ O(u−4 ) with u = x−1 gives the expansion of log qt ; the second term is smaller by a factor 2t

e−2x/t . In (24) the term qt (x) dominates: the nearest competitor qt (4 − x) has support-edge distance 3 − x, and (3 − x)2 − (x − 1)2 = 8 − 4x ≥ 8 − 4 max K > 0, so its ratio to qt (x) is O(e−(8−4x)/2t ); remaining images are farther by at least distance 3. Differentiating in x, the exponentially small corrections are absorbed in the errors, and v = − 12 ∂x log pt . □ Theorem 5.9 (Failure of transport at the data endpoint). Let v = − 21 ∂x log pt with pt as in (24), and let V := {1 < |x| < 2}. RTR (i) For every compact K ⊂ V , 0 K |v| dx dt = +∞: the interior part of (F2) fails. The collar conditions fail as well: the 1/t blow-up persists up to the walls, and near x = ±2 one has [∂x2 log pt ]+ ≍ t−2 , so (F3) also fails there as t ↓ 0. Hypotheses (F0)–(F1) hold, and tangency (F4) holds for every t > 0. (ii) For every x0 ∈ V there is no absolutely continuous Z : [0, T ] → [−2, 2] with Z(0) = x0 Rt and Z(t) = x0 + 0 v(Z(s), s) ds. Since L1 (V ) > 0, no regular Lagrangian flow on Ω̄ exists from time 0. (iii) By contrast, Z ≡ 2 and Z ≡ −2 are integral curves from the boundary points; for every x0 ∈ (−1, 1) the quantile flow Z(t, x0 ) = Ft−1 (F0 (x0 )) is the unique integral curve, with ∂x Z(t, x) = p0 (x)/pt (Z(t, x)), and the transport identity Z(t, ·)# (p0 L1 ) = pt L1 holds. (iv) On [δ, T ], 5.1 applies, with constants of order δ −1 . Rt Proof. (i) is Lemma 5.8 together with 0 0 dt/t = ∞; positivity of pt gives (F1); the claim on [∂x2 log pt ]+ near the walls follows from the fact that at x = 2 the two dominant images make pt even about 2 with a local minimum there, and ∂x2 log pt (2) = p′′t (2)/pt (2) = qt′′ (2)/qt (2) (1 + o(1)) ≍ t−2 ; tangency R xat the walls is the Neumann symmetry noted above. (ii) Set Ft (x) := −2 pt . The flux vanishes at x = −2, so Lemma 2.5 applies on (−2, 2), and t 7→ Ft (Z(t)) is constant along any integral curve on (0, T ], where v is smooth. The new feature of the bounded domain: Ft : [−2, 2] → [0, 1] is onto, and for t > 0 strictly increasing, so the value 1 is attained — exactly at x = 2. Let Z be an integral curve with Z(0) = x0 ∈ (1, 2). Since ∥Ft − F0 ∥∞ ≤ ∥pt − p0 ∥L1 → 0 and Z is continuous at 0, Ft (Z(t)) ≡ q := lim Ft (Z(t)) = F0 (x0 ) = 1. t↓0

For t > 0, Ft (Z(t)) = 1 forces Z(t) = 2; continuity at t = 0 then gives x0 = 2, a contradiction: an integral curve from the vacuum would have to jump instantaneously to the reflecting wall. The case x0 ∈ (−2, −1) is symmetric. Rt (iii) Z ≡ 2 satisfies the integral equation since v(2, t) = 0 for t > 0 and 0 |v(2, s)| ds = 0. For x0 ∈ (−1, 1): q := F0 (x0 ) ∈ (0, 1), Z(t, x0 ) := Ft−1 (q) is well defined, Ż = v(Z, t) by Lemma 2.5, Z(t, x0 ) → x0 as t ↓ 0 by uniform convergence of Ft and strict monotonicity, uniqueness holds because v is locally Lipschitz on (−2, 2)×(0, T ] and the first integral pins the curve; differentiating Ft (Z) = F0 (x) gives the Jacobian identity and the transport identity. The image corrections in (24) are O(e−c/t ) on the relevant compacts and affect no step. (iv) is 5.1; the Tweedie identity applied to the periodised initial law gives |v| ≤ C/δ and ∥∂x v∥∞ ≤ C/δ 2 on [δ, T ]. □ Remark 5.10 (The boundary is innocent). The only genuinely boundary-related hypothesis of Theorem A — tangency — holds here at every positive time, and the boundary points

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carry honest integral curves. What fails is interior time-integrability, through the 1/t blow-up of the score in the vacuum: the same data mechanism as in the whole-space counterexample of [10]. The comparison is instructive: on R, trajectories from the vacuum are excluded because the extreme quantile level is never attained; on [−2, 2] it is attained, at the wall, and trajectories are excluded because reaching it would violate continuity. Reflection changes the mechanism of the contradiction but not the conclusion. 6. Rigidity: constraints on boundary mechanisms Three rigidity results delimit the possible failures of deterministic confined transport, and show that under their hypotheses the structure realised in Theorem B is forced. They do not amount to a classification: they exclude the failure of invariance in one dimension, the absence of confined characteristics whenever the flux is integrable, and the failure of tangency wherever the velocity is bounded with a nondegenerate boundary vacuum. Outside these hypotheses — in particular for low-regularity fields in higher dimension — other boundary mechanisms are not excluded. Proposition 6.1 (One-dimensional rigidity). Let Ω = (a, b) and let (pt ) be a no-flux density evolution with pt > 0 a.e. in Ω for every t ∈ [0, T ], pt v ∈ L1loc , and t 7→ pt L1 narrowly continuous. Then the quantile flow Z(t, x) := Ft−1 (F0 (x)) is well defined for a.e. x, takes values in [a, b], transports p0 L1 onto pt L1 , and its trajectories are integral curves of v wherever Lemma 2.5 applies. Invariance can never fail in d = 1: mass conservation and no-flux pin Ft (a) = 0, Ft (b) = 1, and every quantile level q ∈ (0, 1) is attained inside [a, b] at every time. The only Lagrangian pathologies available in one dimension are those of Theorem 5.9 — nonexistence of curves from an initial vacuum, or unbounded compression where pt degenerates — never an exit through the boundary. Proof. Ft : [a, b] → [0, 1] is continuous, nondecreasing, onto by mass conservation and no-flux, and strictly increasing since pt > 0 a.e.; the rest follows from Lemma 2.5, whose flux-vanishing hypothesis at a is the no-flux condition, and from narrow continuity. □ Proposition 6.2 (Superposition confinement). Let Ω be a bounded Lipschitz domain and (p, J) a no-flux weak solution with J ∈ L1 ([0, T ] × Ω), and assume that the flux vanishes on the vacuum, (25)

J =0

Ld+1 -a.e. on {p = 0},

setting v := J/p there (say v := 0); (25) holds in particular whenever pt > 0 a.e. for a.e. t,
as under (F1). Then there exists η ∈ P C([0, T ]; Rd ) concentrated on integral curves of v which remain in Ω̄ for all t, with (et )# η = pt Ld for every t. Proof. As in Step 3 of the proof of Theorem A, the zero extensions solve the continuity equation on Rd . Hypothesis (25) gives p|v| = |J| Ld+1 -a.e. — on {p > 0} by definition of v, and on {p = 0} because both sides vanish — so the integrability hypothesis of Theorem 2.4 RTR R T R |v| p dx dt ≤ 0 Ω |J| < ∞. The superposition η has time-marginals carried by reads 0 1+|x| Ω̄, and curves being continuous, η-a.e. curve lies in Ω̄ for all rational, hence all, times. □ Remark 6.3. Hypothesis (25) cannot be omitted: Definition 2.1 permits pt to vanish on a set of positive measure, where v = J/p is undefined and the identity p|v| = |J| — the only point at which J ∈ L1 is converted into the weighted bound required by Theorem 2.4 — has no meaning. In all applications made here the hypothesis is automatic, the density being strictly positive in the interior.

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Thus, whenever the total flux is integrable, invariant characteristics exist from p0 -a.e. starting point: a counterexample can only target the flow structure — uniqueness, measurable selection, or compressibility — never the existence of confined integral curves. This is precisely the situation of Theorem B. Lemma 6.4 (Taylor rigidity at a boundary vacuum). Let Γ ⊂ ∂Ω be a relatively open C 1 boundary patch, U a neighbourhood of Γ, and suppose that on (Ω ∩ U ) × (t1 , t2 ): J ∈ C 1 up to Γ; p ∈ C 1 up to Γ with p|Γ ≡ 0 and p(x) ≥ c dist(x, Γ); and v = J/p ∈ L∞ . If the no-flux condition holds on Γ, then v(x, t) · ν → 0 as x → Γ, locally uniformly: tangency cannot fail where the velocity is bounded. Proof. On Γ: J · ν = 0 (no-flux) and |Jτ | = p|vτ | ≤ ∥v∥∞ p → 0, so J|Γ = 0; since J ∈ C 1 and Jτ vanishes identically on Γ, all tangential derivatives of Jτ vanish there, i.e. ∇τ · Jτ |Γ = 0. Moreover p|Γ ≡ 0 on a time interval gives ∂t p|Γ = 0, and the continuity equation yields ∂ν (J · ν)|Γ = −∂t p|Γ − ∇τ · Jτ |Γ = 0. Hence J · ν = o(dist(x, Γ)), and the nondegeneracy of p gives v · ν → 0. □ Lemma 6.4 forces the structure (11) of any boundary-caused example: the tangential flux must not vanish at the wall. The price is unavoidable, and quantifies the entanglement of the boundary hypotheses of Theorem A. Corollary 6.5 (Failure of tangency forces non-integrability). Let Γ ⊂ ∂Ω be a relatively open C 1,1 patch, let p and J be of class C 1 in space and time up to Γ on a time interval, with J · ν = 0 on Γ, and suppose p vanishes on Γ nondegenerately: p(x) ≥ c dist(x, Γ) with c > 0. Let x0 ∈ Γ and ν0 := ν(x0 ). Fix t0 in the stated time interval. If v(x, t0 ) · ν0 ̸−→ 0

as Ω ∋ x → x0

(the approach being unrestricted ), then v(., t0 ) ∈ / L1 on any spatial collar of x0 . Proof. We argue invariantly. Fix x0 ∈ Γ, write ν0 := ν(x0 ) and let {τ1 , . . . , τd−1 } be an orthonormal basis of the tangent space ν0⊥ ; only the single vector ν0 is ever differentiated against. No derivative of the normal field is required; the C 1,1 assumption is used only to obtain the tubular nearest-point projection. The curved boundary enters only through tangential differentiation of functions defined on Ω̄, which for a C 1 patch is well defined without any chart: for τ ∈ ν0⊥ we set ∂τ w(x0 ) := (w ◦ γ)′ (0) for any C 1 curve γ in Γ with γ(0) = x0 , γ ′ (0) = τ , which agrees with the ambient derivative τ · ∇w(x0 ) whenever w is C 1 up to Γ and is therefore independent of the curve. This is the only property of Γ used below, and it is why no flattening — which would not preserve the divergence, and would require second-order regularity of ∂Ω to control the error — is needed. We also record, for later use, that the C 1 regularity of p together with p|Γ ≡ 0 gives the upper bound p(x) ≤ ∥∇p∥∞ dist(x, ∂Ω) near x0 , so that the nondegeneracy hypothesis makes p ≍ dist there. Step 1: ∇ · J = 0 on Γ. Since p|Γ ≡ 0 throughout a time interval, ∂t p = 0 on Γ; the continuity equation holds in Ω and both of its terms extend continuously to Γ, so ∇· J = −∂t p = 0 there. Step 2: One of the following holds: (A) J vanishes identically on some relatively open Γ′ ⊂ Γ containing x0 ; or (B) every relative neighbourhood of x0 in Γ contains a point at which J ̸= 0. We show that (A) implies v(x) · ν0 → 0 as Ω ∋ x → x0 , and that (B) implies v ∈ / L1 on every collar of x0 . The corollary follows: if the limit fails, (A) is excluded by the first implication,

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so (B) holds and the second applies. Arguing through the dichotomy, rather than through the value of a directional limit, is what allows the hypothesis to be the unrestricted limit; a computation along the inward normal ray alone would prove only the weaker statement in which that particular limit is assumed nonzero. Step 3: (A) implies tangency at x0 . Let J ≡ 0 on Γ′ . For x′ ∈ Γ′ and τ tangent at x′ , differentiating along a C 1 curve in Γ′ gives ∂τ J(x′ ) = 0; hence, with an orthonormal tangent frame {τi′ } at x′ and ν ′ := ν(x′ ), Step 1 yields (26)

ν ′⊤ (∇J)(x′ ) ν ′ = tr(∇J)(x′ ) −

d−1 X

τi′ · ∂τi′ J (x′ ) = 0,

i=1

using τ ⊤ (∇J)τ = τ · ∂τ J.

No curvature term appears here: the second fundamental form would enter multiplied by J · ν, which vanishes on Γ. Now let π be a nearest-point projection onto ∂Ω, defined for x near x0 , so that π(x) ∈ Γ′ and x − π(x) = −d(x) ν(π(x)) with d(x) := dist(x, ∂Ω). Since ∇J is uniformly continuous near x0 and J(π(x)) = 0,
J(x) = −d(x) (∇J)(π(x)) ν(π(x)) + o d(x) , hence
J(x) · ν0 = −d(x) ν0⊤ (∇J)(π(x)) ν(π(x)) + o d(x) . As x → x0 we have π(x) → x0 and, Γ being C 1 , ν(π(x)) → ν0 ; by continuity of ∇J and (26) at x′ = x0 , the bracketed factor tends to ν0⊤ (∇J)(x0 )ν0 = 0. Therefore J(x) · ν0 = o(d(x)), and the nondegeneracy p ≥ c d gives o(d(x)) |J(x) · ν0 | ≤ −→ 0, v(x) · ν0 = p(x) c d(x) the approach being unrestricted. Step 4: (B) implies non-integrability. Fix any collar of x0 and choose, inside it, a point x1 ∈ Γ with J(x1 ) ̸= 0. By continuity there are η > 0 and r > 0 with |J| ≥ η on Br (x1 ) ∩ Ω, and by the upper bound recorded above p ≤ ∥∇p∥∞ d there, whence η |J| ≥ on Br (x1 ) ∩ Ω. p ∥∇p∥∞ d(·) Rr / L1 (Br (x1 ) ∩ Ω) and a fortiori Integrating in the normal direction, 0 s−1 ds = ∞, so v ∈ 1 v∈ / L on the collar. As the collar was arbitrary, this holds for every collar of x0 . □ |v| =

Thus, in the class covered by Lemma 6.4, tangency and the collar integrability of v cannot fail independently: the mechanism that destroys one destroys the other. This is consistent with Remark 3.4(a) and delimits what the example of Section 4 can prove; see Remark 4.9. Remark 6.6 (Two independent mechanisms). Theorem 5.9 exhibits the data mechanism: a vacuum in p0 makes integral curves fail to exist, at t = 0 only, in any dimension including d = 1. Theorem B exhibits the boundary mechanism: a boundary current makes the flow property fail on every time interval — at some time t∗ ∈ (0, T ] for each T > 0, hence arbitrarily close to time zero, though not necessarily at each prescribed time, the orbits being periodic — with all integral curves present — and Proposition 6.1 shows this mechanism is genuinely higher-dimensional, since the tangential direction along the wall is what carries the current. The discrepancy between the density evolution and its deterministic transport representation, identified in [10], therefore has (at least) two independent sources, of which exactly one survives in dimension one.

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7. Discussion and extensions Circling back to the reflected diffusion models discussed in Section 1.1, we can observe that our results provide a rigorous mathematical justification for using the reflection-free probability-flow ODE (2) for simulating the flow. After positive-time regularization and under suitable drift–score regularity, this ODE generates a confined regular Lagrangian flow and transports the same marginals as the reflected diffusion, even though it contains no explicit reflection term. This supports deterministic sampling between positive times and clarifies that the relevant boundary condition is tangency of the full probability-flow velocity, which does not necessarily require a zero normal score at the boundary. Importantly, the result is not restricted to hypercubes or polytopes; Corollary 5.1 applies to general bounded C 2,α domains such as the ones considered by Fishman et al. [18]. The results also identify important limitations of the deterministic sampling method. Equality of the marginal distributions alone does not guarantee a well-posed or stable deterministic sampler (2): singular initial data can obstruct a flow at time zero, and boundary degeneracy can produce unbounded compression even when confined characteristics and exact marginal transport exist. Thus early stopping, control of the combined drift–score field, and correct boundary behavior are genuine mathematical requirements rather than technical afterthoughts. These effects illustrate the gap between Eulerian and Lagrangian well-posedness, which can be relevant in applications. Proposition 2.6 shows that the no-flux Fokker–Planck density evolution is unique for bounded measurable forward drifts, whereas construction of a regular Lagrangian flow for the probability-flow ODE requires additional regularity, divergence, and boundary-trace assumptions on the combined drift–score field. Thus the density flow and its time reversal are uniquely determined under weaker assumptions than those needed to justify deterministic probability-flow sampling via (2). From a theoretical perspective, many ramifications of the problems studied here remain to be explored. A natural extension would be to explore the case of obliquely reflected diffusions. The Eulerian object is then, in general, a measure pair –interior measure and boundary measure [21, 23]– and the tangential component of the reflection field induces a surface flux along ∂Ω, so that tangency of the probability-flow velocity fails at the level of the boundary condition itself. Whether a deterministic probability flow exists in this setting is not clear. In this regard, the compression analysis of Theorem B provides a natural model for the interaction of a tangential wall flux with Lagrangian transport. Acknowledgements. The author thanks Luigi Ambrosio for his beautiful lectures at the Spring School on Mathematics of Random Systems (Pisa 2025) and RenYuan Xu for enlightening discussions on generative models.

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